Derivatives of trigonometric functions practice problems
sin, cos, tan, sec, csc and cot, alone and in combination. 233 problems with worked solutions; in 191 of them every
equation is proved by a computer algebra system.
20 worked problems
Try each one before opening the solution. Every line is labeled with what checked it; 16 of these also have a second solution, written separately and checked the same way. How we verify
algebraalgebraCombine the terms into a single fraction. Distribute the negative sign and the constant 2.✓ Proved
\[ = 2 \csc{\left(x + 1 \right)} \]
algebrasimplifyFactor out csc(x + 1) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{2}{\sin{\left(x + 1 \right)}} \)
Top: the graph of f = -2log(cot(x + 1) + csc(x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2(-cot(x + 1)² - cot(x + 1)·csc(x + 1) - 1)/(cot(x + 1) + csc(x + 1)).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
derivativealgebraDifferentiate the inner function x+1 to get 1. Simplify the numerator by combining terms.✓ Proved
\[ = 2 \csc{\left(x + 1 \right)} \]
algebrasimplifyalgebraFactor out -csc(x+1) from the numerator. Cancel the common factor (cot(x+1) + csc(x+1)) in numerator and denominator. Simplify the signs to get the final result.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
algebraCombine the terms over a common denominator.✓ Proved
\[ = \csc{\left(x - 3 \right)} \]
algebrasimplifyFactor out csc(x - 3) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\sin{\left(x - 3 \right)}} \)
Top: the graph of f = -log(cot(x - 3) + csc(x - 3)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(-cot(x - 3)² - cot(x - 3)·csc(x - 3) - 1)/(cot(x - 3) + csc(x - 3)).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
constant-multipleEvaluate the derivative of the inner linear term (x-3), which is 1.✓ Proved
\[ = \csc{\left(x - 3 \right)} \]
algebrasimplifyFactor out -csc(x-3) from the numerator terms. Cancel the common factor (cot(x-3) + csc(x-3)) in the numerator and denominator.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
algebraDistribute the negative sign and the constant.✓ Proved
\[ = 5 \csc{\left(x - 1 \right)} \]
algebrasimplifyFactor out csc(x - 1) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\sin{\left(x - 1 \right)}} \)
Top: the graph of f = -5log(cot(x - 1) + csc(x - 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -5(-cot(x - 1)² - cot(x - 1)·csc(x - 1) - 1)/(cot(x - 1) + csc(x - 1)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Differentiate \( \displaystyle f(x) = \frac{2 \ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \).
\[ \frac{d}{d x} \frac{2 \ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{2 \frac{d}{d x} \ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{2 \frac{d}{d x} \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = \frac{2 \left(\frac{d}{d x} \tan{\left(5 x \right)} + \frac{d}{d x} \sec{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = \frac{2 \left(5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} + 5 \sec^{2}{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
chainApply the chain rule to the trigonometric terms.✓ Proved
\[ = \frac{2 \left(\tan{\left(5 x \right)} \sec{\left(5 x \right)} + \sec^{2}{\left(5 x \right)}\right)}{\tan{\left(5 x \right)} + \sec{\left(5 x \right)}} \]
constant-multiplealgebraFactor out the constant 5. Cancel the 5 in the numerator and denominator.✓ Proved
\[ = \frac{2 \tan{\left(5 x \right)} \sec{\left(5 x \right)} + 2 \sec^{2}{\left(5 x \right)}}{\tan{\left(5 x \right)} + \sec{\left(5 x \right)}} \]
algebraCombine the terms into a single fraction.✓ Proved
\[ = 2 \sec{\left(5 x \right)} \]
algebrasimplifyFactor out sec(5*x) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{2}{\cos{\left(5 x \right)}} \)
Top: the graph of f = 2log(tan(5x) + sec(5x))/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2(5tan(5x)² + 5tan(5x)·sec(5x) + 5)/(5(tan(5x) + sec(5x))).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{2 \ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]
derivative✓ Proved
\[ = \frac{2 \frac{d}{d x} \ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]
constant-multiple✓ Proved
\[ = \frac{2 \frac{d}{d x} \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
chain✓ Proved
\[ = \frac{2 \left(\frac{d}{d x} \tan{\left(5 x \right)} + \frac{d}{d x} \sec{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
sum✓ Proved
\[ = \frac{2 \left(5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} + 5 \sec^{2}{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
derivative✓ Proved
\[ = \frac{2 \tan{\left(5 x \right)} \sec{\left(5 x \right)} + 2 \sec^{2}{\left(5 x \right)}}{\tan{\left(5 x \right)} + \sec{\left(5 x \right)}} \]
simplify✓ Proved
\[ = 2 \sec{\left(5 x \right)} \]
algebrasimplify✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
chainDifferentiate the inner linear function x + 1.✓ Proved
\[ = \csc{\left(x + 1 \right)} \]
algebraalgebrasimplifyFactor out -csc(x + 1) from the numerator. Cancel the common term in the numerator and denominator. Simplify the remaining expression.✓ Proved
Answer \( \frac{1}{\sin{\left(x + 1 \right)}} \)
Top: the graph of f = -log(cot(x + 1) + csc(x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(-cot(x + 1)² - cot(x + 1)·csc(x + 1) - 1)/(cot(x + 1) + csc(x + 1)).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
derivativesimplifyDifferentiate the inner linear term x + 1 to get 1. Simplify the multiplication by 1.✓ Proved
\[ = \csc{\left(x + 1 \right)} \]
algebrasimplifyFactor out -csc(x+1) from the numerator terms. Cancel the common term (cot(x+1) + csc(x+1)) in numerator and denominator.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \).
\[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]
constant-multipleMove the constant outside the derivative.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
sumDifferentiate the sum inside the derivative.✓ Proved
\[ = \frac{\frac{d}{d x} \tan{\left(5 x \right)} + \frac{d}{d x} \sec{\left(5 x \right)}}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
chainApply the chain rule to each term.✓ Proved
\[ = \frac{5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} + 5 \sec^{2}{\left(5 x \right)}}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
constant-multipleFactor out the constant 5.✓ Proved
\[ = \frac{\tan{\left(5 x \right)} \sec{\left(5 x \right)} + \sec^{2}{\left(5 x \right)}}{\tan{\left(5 x \right)} + \sec{\left(5 x \right)}} \]
algebrasimplifySimplify the expression by canceling 5 and 1/5. Simplify the fraction.✓ Proved
\[ = \sec{\left(5 x \right)} \]
algebrasimplifyFactor out sec(5*x) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\cos{\left(5 x \right)}} \)
Top: the graph of f = log(tan(5x) + sec(5x))/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (5tan(5x)² + 5tan(5x)·sec(5x) + 5)/(5(tan(5x) + sec(5x))).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]
derivative✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)} \right)}}{5} \]
constant-multiple✓ Proved
\[ = \frac{\frac{d}{d x} \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
logarithmic✓ Proved
\[ = \frac{\frac{d}{d x} \tan{\left(5 x \right)} + \frac{d}{d x} \sec{\left(5 x \right)}}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
sum✓ Proved
\[ = \frac{5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} + 5 \sec^{2}{\left(5 x \right)}}{5 \left(\tan{\left(5 x \right)} + \sec{\left(5 x \right)}\right)} \]
chain✓ Proved
\[ = \frac{\tan{\left(5 x \right)} \sec{\left(5 x \right)} + \sec^{2}{\left(5 x \right)}}{\tan{\left(5 x \right)} + \sec{\left(5 x \right)}} \]
algebrasimplifyFactor out the common 5✓ Proved
\[ = \sec{\left(5 x \right)} \]
simplify✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
algebraCombine the terms into a single fraction.✓ Proved
\[ = 5 \csc{\left(x \right)} \]
algebrasimplifyFactor out csc(x) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\sin{\left(x \right)}} \)
Top: the graph of f = 5log(-cot(x) + csc(x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5(cot(x)² - cot(x)·csc(x) + 1)/(-cot(x) + csc(x)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
constant-multiplePull out the constant -2/3✓ Proved
\[ = - \frac{2 \frac{d}{d x} \cos{\left(3 x \right)}}{3 \cos{\left(3 x \right)}} \]
derivativeDerivative of log(u) is u' / u✓ Proved
\[ = \frac{2 \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}} \]
derivativesimplifysimplifyDerivative of cos(3*x) is -sin(3*x)*3 Cancel the negatives and the 3 in numerator and denominator Simplify the fraction✓ Proved
\[ = 2 \tan{\left(3 x \right)} \]
trigsin/cos is tan✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
algebrasimplifyFactor out sec(x) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{2}{\cos{\left(x \right)}} \)
Top: the graph of f = 2log(tan(x) + sec(x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2(tan(x)² + tan(x)·sec(x) + 1)/(tan(x) + sec(x)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
derivativeDerivative of tan(x) is sec(x)**2; derivative of sec(x) is sec(x)*tan(x).✓ Proved
\[ = 2 \sec{\left(x \right)} \]
algebrasimplifyFactor sec(x) from the numerator. Cancel (sec(x)+tan(x)) with (tan(x)+sec(x)).✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{\frac{d}{d x} \cos{\left(2 x + 1 \right)}}{2 \cos{\left(2 x + 1 \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = \frac{\sin{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x + 1\right)}{2 \cos{\left(2 x + 1 \right)}} \]
chainApply the chain rule for the cosine function.✓ Proved
\[ = \frac{\sin{\left(2 x + 1 \right)}}{\cos{\left(2 x + 1 \right)}} \]
derivativeconstant-multipleDifferentiate the innermost linear function. Simplify the constant factors.✓ Proved
\[ = \tan{\left(2 x + 1 \right)} \]
algebrasimplifyUse the identity sec(x) = 1/cos(x) and simplify signs. Final simplification.✓ Proved
Answer \( \tan{\left(2 x + 1 \right)} \)
Top: the graph of f = -log(cos(2x + 1))/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sin(2x + 1)/cos(2x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ = - \frac{\frac{d}{d x} \cos{\left(2 x + 1 \right)}}{2 \cos{\left(2 x + 1 \right)}} \]
logarithmic✓ Proved
\[ = \frac{\sin{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x + 1\right)}{2 \cos{\left(2 x + 1 \right)}} \]
trig✓ Proved
\[ = \frac{\sin{\left(2 x + 1 \right)}}{\cos{\left(2 x + 1 \right)}} \]
constantsimplify✓ Proved
\[ = \tan{\left(2 x + 1 \right)} \]
simplify✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{\frac{d}{d x} \cos{\left(3 x - 1 \right)}}{3 \cos{\left(3 x - 1 \right)}} \]
logarithmicApply the chain rule for the logarithm.✓ Proved
\[ = \frac{\sin{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{3 \cos{\left(3 x - 1 \right)}} \]
trigApply the chain rule for the cosine function.✓ Proved
\[ = \frac{\sin{\left(3 x - 1 \right)}}{\cos{\left(3 x - 1 \right)}} \]
derivativeconstant-multiplealgebraDifferentiate the inner linear function. Simplify the constant factors. Multiply the terms together.✓ Proved
\[ = \tan{\left(3 x - 1 \right)} \]
simplifyUse the trigonometric identity tan(u) = sin(u)/cos(u).✓ Proved
Answer \( \tan{\left(3 x - 1 \right)} \)
Top: the graph of f = -log(cos(3x - 1))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sin(3x - 1)/cos(3x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ = - \frac{\frac{d}{d x} \cos{\left(3 x - 1 \right)}}{3 \cos{\left(3 x - 1 \right)}} \]
productPull out the constant -1/3 and apply derivative of log✓ Proved
\[ = \frac{\sin{\left(3 x - 1 \right)}}{\cos{\left(3 x - 1 \right)}} \]
chainDerivative of cos(3*x - 1) is -sin(3*x - 1)*3✓ Proved
\[ = \tan{\left(3 x - 1 \right)} \]
simplifySimplify the constants and the ratio of sin to cos✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Differentiate \( \displaystyle f(x) = \ln{\left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \).
\[ \frac{d}{d x} \ln{\left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
chainApply the chain rule for the natural logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \left(- \cot{\left(3 x \right)}\right) + \frac{d}{d x} \csc{\left(3 x \right)}}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
sumApply the sum rule to the derivative of the inner expression.✓ Proved
\[ = \frac{- \frac{d}{d x} \cot{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
constantFactor out the negative sign from the first term.✓ Proved
\[ = \frac{- 3 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + 3 \csc^{2}{\left(3 x \right)}}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
trigalgebraDifferentiate the trigonometric functions using the chain rule. Simplify the expression by distributing the negative signs.✓ Proved
\[ = 3 \csc{\left(3 x \right)} \]
algebrasimplifyFactor out the common term 3*csc(3*x) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{3}{\sin{\left(3 x \right)}} \)
Top: the graph of f = log(-cot(3x) + csc(3x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (3cot(3x)² - 3cot(3x)·csc(3x) + 3)/(-cot(3x) + csc(3x)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \ln{\left(- \cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \]
derivative✓ Proved
\[ = \frac{3 \cot^{2}{\left(3 x \right)} - 3 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + 3}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
chain✓ Proved
\[ = \frac{- 3 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + 3 \csc^{2}{\left(3 x \right)}}{- \cot{\left(3 x \right)} + \csc{\left(3 x \right)}} \]
derivative✓ Proved
\[ = 3 \csc{\left(3 x \right)} \]
algebrasimplify✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
chainalgebraApply the chain rule to the inner function x + 1. Simplify the numerator by multiplying out the 1s.✓ Proved
\[ = 3 \sec{\left(x + 1 \right)} \]
algebrasimplifyFactor out the common term sec(x + 1) in the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{3}{\cos{\left(x + 1 \right)}} \)
Top: the graph of f = 3log(tan(x + 1) + sec(x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3(tan(x + 1)² + tan(x + 1)·sec(x + 1) + 1)/(tan(x + 1) + sec(x + 1)).
✓ Nihil obstat Lines: 9 proved, 1 checked by independent solution. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
chainconstantalgebraDifferentiate the inner linear terms. Simplify the derivative of the inner terms. Multiply the fraction by the numerator.✓ Proved
\[ = \sec{\left(x + 1 \right)} \]
algebrasimplifyFactor out sec(x + 1) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\cos{\left(x + 1 \right)}} \)
Top: the graph of f = log(tan(x + 1) + sec(x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (tan(x + 1)² + tan(x + 1)·sec(x + 1) + 1)/(tan(x + 1) + sec(x + 1)).
✓ Nihil obstat Lines: 9 proved, 1 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
trigalgebraalgebraDifferentiate the trigonometric functions. Simplify the terms by multiplying by 1. Combine the fraction into a single expression.≈ Checked numerically
\[ = \sec{\left(x - 1 \right)} \]
algebrasimplifyFactor out the common term sec(x - 1) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\cos{\left(x - 1 \right)}} \)
Top: the graph of f = log(tan(x - 1) + sec(x - 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (tan(x - 1)² + tan(x - 1)·sec(x - 1) + 1)/(tan(x - 1) + sec(x - 1)).
✓ Nihil obstat Lines: 9 proved, 1 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
derivativealgebraDifferentiate the inner linear functions. Combine the terms into a single fraction.✓ Proved
\[ = \sec{\left(x - 3 \right)} \]
algebrasimplifyFactor out sec(x - 3) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\cos{\left(x - 3 \right)}} \)
Top: the graph of f = log(tan(x - 3) + sec(x - 3)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (tan(x - 3)² + tan(x - 3)·sec(x - 3) + 1)/(tan(x - 3) + sec(x - 3)).
✓ Nihil obstat Lines: 8 proved, 1 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \).
\[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \]
constantPull out the constant factor.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} \cot{\left(3 x + 1 \right)} + \frac{d}{d x} \csc{\left(3 x + 1 \right)}}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = - \frac{- \cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right) - \csc^{2}{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x + 1\right)}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]
trigApply the derivatives for cot and csc.✓ Proved
\[ = - \frac{- 3 \cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} - 3 \csc^{2}{\left(3 x + 1 \right)}}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]
chainDifferentiate the inner linear function 3x + 1.✓ Proved
\[ = - \frac{- \cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} - \csc^{2}{\left(3 x + 1 \right)}}{\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}} \]
constant-multiplealgebraFactor out the constant 3. Cancel the 3 and the 1/3.✓ Proved
\[ = \frac{\cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} + \csc^{2}{\left(3 x + 1 \right)}}{\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}} \]
algebraDistribute the negative sign.✓ Proved
\[ = \csc{\left(3 x + 1 \right)} \]
algebrasimplifyFactor out csc(3*x + 1) from the numerator. Simplify the fraction.✓ Proved
Answer \( \frac{1}{\sin{\left(3 x + 1 \right)}} \)
Top: the graph of f = -log(cot(3x + 1) + csc(3x + 1))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(-3cot(3x + 1)² - 3cot(3x + 1)·csc(3x + 1) - 3)/(3(cot(3x + 1) + csc(3x + 1))).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(- \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3}\right) \]
derivative✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \]
constant-multiple✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]
chain✓ Proved
\[ = - \frac{\frac{d}{d x} \cot{\left(3 x + 1 \right)} + \frac{d}{d x} \csc{\left(3 x + 1 \right)}}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]
sum✓ Proved
\[ = - \frac{- 3 \cot{\left(3 x + 1 \right)} \csc{\left(3 x + 1 \right)} - 3 \csc^{2}{\left(3 x + 1 \right)}}{3 \left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)}\right)} \]
sum✓ Proved
\[ = \csc{\left(3 x + 1 \right)} \]
algebrasimplifysimplifyFactor -3*csc from the sum✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Differentiate \( \displaystyle f(x) = - \frac{2 \ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5} \).
\[ \frac{d}{d x} \left(- \frac{2 \ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5}\right) \]
constant-multipleStart with the derivative of the function.✓ Proved
\[ = - \frac{2 \frac{d}{d x} \ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{2 \frac{d}{d x} \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = - \frac{2 \left(\frac{d}{d x} \cot{\left(5 x - 3 \right)} + \frac{d}{d x} \csc{\left(5 x - 3 \right)}\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = - \frac{2 \left(- \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right) - \csc^{2}{\left(5 x - 3 \right)} \frac{d}{d x} \left(5 x - 3\right)\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]
trigApply the derivatives of cotangent and cosecant.✓ Proved
\[ = - \frac{2 \left(- 5 \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} - 5 \csc^{2}{\left(5 x - 3 \right)}\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]
chainDifferentiate the inner linear function 5x - 3.✓ Proved
\[ = - \frac{2 \left(- \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} - \csc^{2}{\left(5 x - 3 \right)}\right)}{\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}} \]
constant-multiplealgebraFactor out the common constant 5. Multiply the constants -2/5 and 5.✓ Proved
\[ = \frac{2 \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} + 2 \csc^{2}{\left(5 x - 3 \right)}}{\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}} \]
algebraDistribute the negative sign and simplify.✓ Proved
\[ = 2 \csc{\left(5 x - 3 \right)} \]
algebrasimplifyFactor out csc(5*x - 3) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{2}{\sin{\left(5 x - 3 \right)}} \)
Top: the graph of f = -2log(cot(5x - 3) + csc(5x - 3))/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2(-5cot(5x - 3)² - 5cot(5x - 3)·csc(5x - 3) - 5)/(5(cot(5x - 3) + csc(5x - 3))).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(- \frac{2 \ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5}\right) \]
derivative✓ Proved
\[ = - \frac{2 \frac{d}{d x} \ln{\left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)} \right)}}{5} \]
constant-multiple✓ Proved
\[ = - \frac{2 \frac{d}{d x} \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]
chain✓ Proved
\[ = - \frac{2 \left(\frac{d}{d x} \cot{\left(5 x - 3 \right)} + \frac{d}{d x} \csc{\left(5 x - 3 \right)}\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]
sum✓ Proved
\[ = - \frac{2 \left(- 5 \csc^{2}{\left(5 x - 3 \right)} + \frac{d}{d x} \csc{\left(5 x - 3 \right)}\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]
derivative✓ Proved
\[ = - \frac{2 \left(- 5 \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} - 5 \csc^{2}{\left(5 x - 3 \right)}\right)}{5 \left(\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}\right)} \]
derivativealgebraFactor out -5✓ Proved
\[ = \frac{2 \cot{\left(5 x - 3 \right)} \csc{\left(5 x - 3 \right)} + 2 \csc^{2}{\left(5 x - 3 \right)}}{\cot{\left(5 x - 3 \right)} + \csc{\left(5 x - 3 \right)}} \]
simplify✓ Proved
\[ = 2 \csc{\left(5 x - 3 \right)} \]
algebrasimplifyFactor csc(5*x-3) from numerator Cancel identical factors✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \).
\[ \frac{d}{d x} \frac{\ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \]
constant-multiplePull out the constant factor 1/3.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \]
chainApply the chain rule to the natural logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = \frac{\frac{d}{d x} \cot{\left(3 x - 1 \right)} + \frac{d}{d x} \csc{\left(3 x - 1 \right)}}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]
trigApply the derivative formulas for cotangent and cosecant.✓ Proved
\[ = \frac{- \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right) - \csc^{2}{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]
derivativeEvaluate the derivatives of the inner linear functions.✓ Proved
\[ = \frac{- 3 \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} - 3 \csc^{2}{\left(3 x - 1 \right)}}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]
constant-multipleFactor out the common 3 from the derivative terms.✓ Proved
\[ = \frac{- \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} - \csc^{2}{\left(3 x - 1 \right)}}{\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}} \]
simplifyalgebraCancel the 3 and 1/3. Factor out -csc(3*x - 1) from the numerator.✓ Proved
\[ = - \csc{\left(3 x - 1 \right)} \]
simplifysimplifyCancel the common term in the numerator and denominator. Final simplified result.✓ Proved
Answer \( - \frac{1}{\sin{\left(3 x - 1 \right)}} \)
Top: the graph of f = log(cot(3x - 1) + csc(3x - 1))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (-3cot(3x - 1)² - 3cot(3x - 1)·csc(3x - 1) - 3)/(3(cot(3x - 1) + csc(3x - 1))).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \]
derivativeStart with the derivative of the given function.Not checked
\[ = \frac{\frac{d}{d x} \ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \]
constant-multipleFactor out the constant multiplier 1/3.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]
chainApply the chain rule for the natural logarithm: d/dx log(u) = u'/u.✓ Proved
\[ = \frac{\frac{d}{d x} \cot{\left(3 x - 1 \right)} + \frac{d}{d x} \csc{\left(3 x - 1 \right)}}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]
sumApply the sum rule to the derivative of the argument.✓ Proved
\[ = \frac{- \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right) - \csc^{2}{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]
chainDifferentiate cot(u) and csc(u) using their standard derivatives and the chain rule for the inner function 3x-1.✓ Proved
\[ = \frac{- 3 \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} - 3 \csc^{2}{\left(3 x - 1 \right)}}{3 \left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}\right)} \]
chainalgebraCompute the derivative of the inner function 3x-1, which is 3. Simplify the multiplication by the constant 3.✓ Proved
\[ = \frac{- \cot{\left(3 x - 1 \right)} \csc{\left(3 x - 1 \right)} - \csc^{2}{\left(3 x - 1 \right)}}{\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)}} \]
simplifyCancel the factor of 3 in the numerator and denominator.✓ Proved
\[ = - \csc{\left(3 x - 1 \right)} \]
algebrasimplifyFactor out -csc(3x-1) from the numerator. Cancel the common term (cot(3x-1) + csc(3x-1)) in the numerator and denominator.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)} \).
\[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)} \]
constantPull out the constant factor -1.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}\right)}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} \cot{\left(3 x + 2 \right)} + \frac{d}{d x} \csc{\left(3 x + 2 \right)}}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]
sumApply the sum rule to the inner terms.✓ Proved
\[ = - \frac{- \cot{\left(3 x + 2 \right)} \csc{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right) - \csc^{2}{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right)}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]
trigDifferentiate the trigonometric functions.✓ Proved
\[ = - \frac{- 3 \cot{\left(3 x + 2 \right)} \csc{\left(3 x + 2 \right)} - 3 \csc^{2}{\left(3 x + 2 \right)}}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]
derivativealgebraDifferentiate the linear inner function 3x + 2. Factor out -3 from the derivative term.✓ Proved
\[ = \frac{3 \cot{\left(3 x + 2 \right)} \csc{\left(3 x + 2 \right)} + 3 \csc^{2}{\left(3 x + 2 \right)}}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]
algebraSimplify the signs and the fraction.✓ Proved
\[ = 3 \csc{\left(3 x + 2 \right)} \]
algebrasimplifyFactor out csc(3*x + 2) in the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{3}{\sin{\left(3 x + 2 \right)}} \)
Top: the graph of f = -log(cot(3x + 2) + csc(3x + 2)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(-3cot(3x + 2)² - 3cot(3x + 2)·csc(3x + 2) - 3)/(cot(3x + 2) + csc(3x + 2)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)}\right) \]
derivative✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}\right)}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]
chainApply chain rule to -log(u)✓ Proved
\[ = - \frac{- 3 \cot{\left(3 x + 2 \right)} \csc{\left(3 x + 2 \right)} - 3 \csc^{2}{\left(3 x + 2 \right)}}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]
derivativeDifferentiate cot and csc inside✓ Proved
\[ = \frac{3 \cot{\left(3 x + 2 \right)} \csc{\left(3 x + 2 \right)} + 3 \csc^{2}{\left(3 x + 2 \right)}}{\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)}} \]
simplifyCancel the minus signs✓ Proved
\[ = 3 \csc{\left(3 x + 2 \right)} \]
simplifyFactor csc and cancel the common sum✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
More derivatives of trigonometric functions practice