Derivatives of polynomials, roots and powers of x. 313 problems with worked solutions; in 313 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; 20 of these also have a second solution, written separately and checked the same way. How we verify
Differentiate \( \displaystyle f(x) = 12 x \left(x^{2} + 3 x + 3\right) \).
\[ \frac{d}{d x} 12 x \left(x^{2} + 3 x + 3\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 12 \frac{d}{d x} x \left(x^{2} + 3 x + 3\right) \]
constant-multiplePull out the constant factor 12.✓ Proved
\[ = 12 x \frac{d}{d x} \left(x^{2} + 3 x + 3\right) + 12 \left(x^{2} + 3 x + 3\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = 12 x^{2} + 12 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 3 x + \frac{d}{d x} x^{2}\right) + 36 x + 36 \]
sumApply the sum rule to the terms inside the parentheses.✓ Proved
\[ = 12 x^{2} + 12 x \left(2 x + 3\right) + 36 x + 36 \]
derivativealgebraDifferentiate each term individually. Simplify the expression inside the parentheses.✓ Proved
\[ = 36 x^{2} + 72 x + 36 \]
algebraalgebrasimplifyDistribute x into the parentheses. Combine like terms. Distribute the 12 to get the final simplified form.✓ Proved
Answer \( 36 \left(x + 1\right)^{2} \)
Top: the graph of f = 12x·(x² + 3x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 12x² + 12x·(2x + 3) + 36x + 36.
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} 12 x \left(x^{2} + 3 x + 3\right) \]
derivativeStart by writing the derivative of the given function with respect to x.Not checked
\[ = 12 \frac{d}{d x} x \left(x^{2} + 3 x + 3\right) \]
constant-multipleFactor out the constant multiplier 12 using the constant multiple rule.✓ Proved
\[ = 12 x \frac{d}{d x} \left(x^{2} + 3 x + 3\right) + 12 \left(x^{2} + 3 x + 3\right) \frac{d}{d x} x \]
productApply the product rule to the product of x and (x^2 + 3x + 3).✓ Proved
\[ = 12 x^{2} + 12 x \frac{d}{d x} \left(x^{2} + 3 x + 3\right) + 36 x + 36 \]
powerDifferentiate x with respect to x to get 1.✓ Proved
\[ = 12 x^{2} + 12 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 3 x + \frac{d}{d x} x^{2}\right) + 36 x + 36 \]
sumApply the sum rule to differentiate the polynomial inside the parentheses.✓ Proved
\[ = 12 x^{2} + 12 x \left(2 x + 3\right) + 36 x + 36 \]
powerDifferentiate x^2 to get 2x, 3x to get 3, and the constant 3 to get 0.✓ Proved
\[ = 36 x^{2} + 72 x + 36 \]
algebraalgebraalgebraDistribute x into the terms (2x + 3) to get 2x^2 + 3x. Combine like terms: x^2 + 2x^2 = 3x^2 and 3x + 3x = 6x. Distribute the constant 12 into the polynomial.✓ 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 \( \displaystyle f(x) = 5 x \left(x^{2} - 9 x + 27\right) \).
\[ \frac{d}{d x} 5 x \left(x^{2} - 9 x + 27\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 5 \frac{d}{d x} x \left(x^{2} - 9 x + 27\right) \]
constant-multiplePull out the constant factor 5.✓ Proved
\[ = 5 x \frac{d}{d x} \left(x^{2} - 9 x + 27\right) + 5 \left(x^{2} - 9 x + 27\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = 5 x^{2} + 5 x \left(\frac{d}{d x} 27 - \frac{d}{d x} 9 x + \frac{d}{d x} x^{2}\right) - 45 x + 135 \]
derivativeDifferentiate each term in the product.✓ Proved
\[ = 5 x^{2} + 5 x \left(2 x - 9\right) - 45 x + 135 \]
derivativealgebraEvaluate the derivatives of the individual terms. Simplify the expression inside the parentheses.✓ Proved
\[ = 15 x^{2} - 90 x + 135 \]
algebraalgebraalgebraDistribute x into the parentheses. Combine like terms. Distribute the 5 to get the final simplified form.✓ Proved
Answer \( 15 \left(x - 3\right)^{2} \)
Top: the graph of f = 5x·(x² - 9x + 27). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x² + 5x·(2x - 9) - 45x + 135.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} 5 x \left(x^{2} - 9 x + 27\right) \]
derivative✓ Proved
\[ = 5 x^{2} + 5 x \left(2 x - 9\right) - 45 x + 135 \]
product✓ Proved
\[ = 15 x^{2} - 90 x + 135 \]
algebraalgebrasimplifyExpanded the products Combined like terms Factored the quadratic inside✓ Proved
\[ = 15 \left(x - 3\right)^{2} \]
simplifySimplified the perfect square✓ 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
powersimplifyDifferentiate each power term Simplify the constants✓ 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
✓ 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
powersimplifyApply power rule to each term Simplify the expression✓ 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 \( \displaystyle f(x) = \frac{\left(5 x + 1\right)^{2}}{5 x + 2} \).
\[ \frac{d}{d x} \frac{\left(5 x + 1\right)^{2}}{5 x + 2} \]
derivativerewriteStart with the derivative of the function. Rewrite the quotient using a negative exponent.✓ Proved
\[ = \left(5 x + 1\right)^{2} \frac{d}{d x} \frac{1}{5 x + 2} + \frac{\frac{d}{d x} \left(5 x + 1\right)^{2}}{5 x + 2} \]
productApply the product rule.✓ Proved
\[ = \left(5 x + 1\right)^{2} \frac{d}{d x} \frac{1}{5 x + 2} + \frac{\left(10 x + 2\right) \frac{d}{d x} \left(5 x + 1\right)}{5 x + 2} \]
powerApply the chain rule to the first term.✓ Proved
\[ = \left(5 x + 1\right)^{2} \frac{d}{d x} \frac{1}{5 x + 2} + \frac{50 x + 10}{5 x + 2} \]
derivativeconstant-multipleDifferentiate the inner function 5*x + 1. Multiply the constants.✓ Proved
\[ = - \frac{\left(5 x + 1\right)^{2} \frac{d}{d x} \left(5 x + 2\right)}{\left(5 x + 2\right)^{2}} + \frac{50 x + 10}{5 x + 2} \]
powerApply the chain rule to the second term.✓ Proved
\[ = - \frac{5 \left(5 x + 1\right)^{2}}{\left(5 x + 2\right)^{2}} + \frac{50 x + 10}{5 x + 2} \]
derivativeconstant-multipleDifferentiate the inner function 5*x + 2. Simplify the signs and constants.✓ Proved
\[ = \frac{- 5 \left(5 x + 1\right)^{2} + \left(5 x + 2\right) \left(50 x + 10\right)}{\left(5 x + 2\right)^{2}} \]
algebraFind a common denominator.✓ Proved
\[ = \frac{125 x^{2} + 100 x + 15}{\left(5 x + 2\right)^{2}} \]
algebraalgebrasimplifyExpand the terms in the numerator. Distribute the constants. Combine like terms to get the final answer.✓ Proved
Answer \( \frac{5 \left(5 x + 1\right) \left(5 x + 3\right)}{\left(5 x + 2\right)^{2}} \)
Top: the graph of f = (5x + 1)²/(5x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -5(5x + 1)²/(5x + 2)² + (50x + 10)/(5x + 2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\left(5 x + 1\right)^{2}}{5 x + 2} \]
derivative✓ Proved
\[ = \frac{- \left(5 x + 1\right)^{2} \frac{d}{d x} \left(5 x + 2\right) + \left(5 x + 2\right) \frac{d}{d x} \left(5 x + 1\right)^{2}}{\left(5 x + 2\right)^{2}} \]
quotient✓ Proved
\[ = \frac{- 5 \left(5 x + 1\right)^{2} + \left(5 x + 2\right) \left(10 x + 2\right) \frac{d}{d x} \left(5 x + 1\right)}{\left(5 x + 2\right)^{2}} \]
product✓ Proved
\[ = \frac{- 5 \left(5 x + 1\right)^{2} + \left(5 x + 2\right) \left(50 x + 10\right)}{\left(5 x + 2\right)^{2}} \]
derivativeconstant-multiple✓ Proved
\[ = \frac{- 5 \left(5 x + 1\right)^{2} + 5 \left(5 x + 2\right) \left(10 x + 2\right)}{\left(5 x + 2\right)^{2}} \]
constant-multiple✓ Proved
\[ = \frac{\left(5 x + 3\right) \left(25 x + 5\right)}{\left(5 x + 2\right)^{2}} \]
algebrasimplifysimplify✓ 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
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = - 25 x^{2} + x \left(- 50 x - 15\right) - 15 x - 3 \]
sumalgebraDifferentiate each term in the sum. Simplify the terms inside the parentheses.✓ Proved
\[ = - 75 x^{2} - 30 x - 3 \]
productalgebrasimplifyDistribute x into the parentheses. Remove parentheses. Combine like terms.✓ Proved
Answer \( - 3 \left(5 x + 1\right)^{2} \)
Top: the graph of f = x·(-25x² - 15x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -25x² + x·(-50x - 15) - 15x - 3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} x \left(- 25 x^{2} - 15 x - 3\right) \]
derivative✓ Proved
\[ = - 25 x^{2} + x \left(- 50 x - 15\right) - 15 x - 3 \]
productApply product rule✓ Proved
\[ = - 75 x^{2} - 30 x - 3 \]
simplifysimplifyCombine like terms Final simplified derivative✓ 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 \( \displaystyle f(x) = 20 x \left(4 x^{2} + 6 x + 3\right) \).
\[ \frac{d}{d x} 20 x \left(4 x^{2} + 6 x + 3\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 20 \frac{d}{d x} x \left(4 x^{2} + 6 x + 3\right) \]
constant-multiplePull out the constant factor 20.✓ Proved
\[ = 20 x \frac{d}{d x} \left(4 x^{2} + 6 x + 3\right) + 20 \left(4 x^{2} + 6 x + 3\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = 80 x^{2} + 20 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 6 x + \frac{d}{d x} 4 x^{2}\right) + 120 x + 60 \]
derivativeDifferentiate each term in the sum.✓ Proved
\[ = 80 x^{2} + 20 x \left(8 x + 6\right) + 120 x + 60 \]
derivativealgebraEvaluate the derivatives of the individual terms. Simplify the expression inside the parentheses.✓ Proved
\[ = 240 x^{2} + 240 x + 60 \]
algebraalgebraalgebraDistribute x into the parentheses. Combine like terms. Distribute the 20 to get the final simplified form.✓ Proved
Answer \( 60 \left(2 x + 1\right)^{2} \)
Top: the graph of f = 20x·(4x² + 6x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 80x² + 20x·(8x + 6) + 120x + 60.
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} 20 x \left(4 x^{2} + 6 x + 3\right) \]
derivativeStart by writing the derivative of the given function with respect to x.Not checked
\[ = 20 \frac{d}{d x} x \left(4 x^{2} + 6 x + 3\right) \]
constant-multipleFactor out the constant coefficient 20 using the constant multiple rule.✓ Proved
\[ = 20 x \frac{d}{d x} \left(4 x^{2} + 6 x + 3\right) + 20 \left(4 x^{2} + 6 x + 3\right) \frac{d}{d x} x \]
productApply the product rule to the term x*(4*x^2 + 6*x + 3).✓ Proved
\[ = 80 x^{2} + 20 x \frac{d}{d x} \left(4 x^{2} + 6 x + 3\right) + 120 x + 60 \]
derivativeDifferentiate x with respect to x to get 1.✓ Proved
\[ = 80 x^{2} + 20 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 6 x + \frac{d}{d x} 4 x^{2}\right) + 120 x + 60 \]
sumApply the sum rule to differentiate the polynomial inside the parentheses.✓ Proved
\[ = 80 x^{2} + 20 x \left(8 x + 6\right) + 120 x + 60 \]
powerDifferentiate each term: 4x^2 becomes 8x, 6x becomes 6, and the constant 3 becomes 0.✓ Proved
\[ = 240 x^{2} + 240 x + 60 \]
algebraalgebraalgebraDistribute x into the parentheses and simplify terms. Combine like terms inside the parentheses: 4x^2 + 8x^2 = 12x^2 and 6x + 6x = 12x. Distribute the 20 to get the final simplified polynomial.✓ 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 \( \displaystyle f(x) = x \left(- 4 x^{2} + 18 x - 27\right) \).
\[ \frac{d}{d x} x \left(- 4 x^{2} + 18 x - 27\right) \]
Start with the derivative of the function.✓ Proved
\[ = x \frac{d}{d x} \left(- 4 x^{2} + 18 x - 27\right) + \left(- 4 x^{2} + 18 x - 27\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = - 4 x^{2} + x \frac{d}{d x} \left(- 4 x^{2} + 18 x - 27\right) + 18 x - 27 \]
derivativeconstant-multipleDifferentiate the first part of the product. Simplify the derivative of x.✓ Proved
\[ = - 4 x^{2} + x \left(- \frac{d}{d x} 27 + \frac{d}{d x} 18 x + \frac{d}{d x} \left(- 4 x^{2}\right)\right) + 18 x - 27 \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = - 4 x^{2} + x \left(18 - 8 x\right) + 18 x - 27 \]
derivativeconstantDifferentiate each term in the sum. The derivative of the constant 27 is 0.✓ Proved
\[ = - 12 x^{2} + 36 x - 27 \]
algebrasimplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( - 12 x^{2} + 36 x - 27 \)
Top: the graph of f = x·(-4x² + 18x - 27). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -4x² + x·(18 - 8x) + 18x - 27.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} x \left(- 4 x^{2} + 18 x - 27\right) \]
derivative✓ Proved
\[ = - 4 x^{2} + x \frac{d}{d x} \left(- 4 x^{2} + 18 x - 27\right) + 18 x - 27 \]
productApply product rule✓ Proved
\[ = - 4 x^{2} + x \left(18 - 8 x\right) + 18 x - 27 \]
derivativeDifferentiate the quadratic✓ Proved
\[ = - 12 x^{2} + 36 x - 27 \]
algebrasimplifyDistribute the x Combine like terms✓ 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 \( \displaystyle f(x) = 5 x \left(25 x^{2} - 15 x + 3\right) \).
\[ \frac{d}{d x} 5 x \left(25 x^{2} - 15 x + 3\right) \]
Start with the derivative of the function.✓ Proved
\[ = 5 \frac{d}{d x} x \left(25 x^{2} - 15 x + 3\right) \]
constant-multiplePull out the constant factor 5.✓ Proved
\[ = 5 x \frac{d}{d x} \left(25 x^{2} - 15 x + 3\right) + 5 \left(25 x^{2} - 15 x + 3\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = 125 x^{2} + 5 x \left(\frac{d}{d x} 3 - \frac{d}{d x} 15 x + \frac{d}{d x} 25 x^{2}\right) - 75 x + 15 \]
algebraDistribute the derivative across the sum.✓ Proved
\[ = 125 x^{2} + 5 x \left(\frac{d}{d x} 25 x^{2} - 15\right) - 75 x + 15 \]
derivativeDifferentiate the individual terms.✓ Proved
\[ = 125 x^{2} + 5 x \left(50 x - 15\right) - 75 x + 15 \]
derivativeEvaluate the derivative of 25*x**2.✓ Proved
\[ = 375 x^{2} - 150 x + 15 \]
algebraalgebraalgebraDistribute x into the parentheses. Combine like terms inside the parentheses. Distribute the constant 5 to get the final answer.✓ Proved
Answer \( 15 \left(5 x - 1\right)^{2} \)
Top: the graph of f = 5x·(25x² - 15x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 125x² + 5x·(50x - 15) - 75x + 15.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} 5 x \left(25 x^{2} - 15 x + 3\right) \]
derivative✓ Proved
\[ = 5 x \frac{d}{d x} \left(25 x^{2} - 15 x + 3\right) + \left(25 x^{2} - 15 x + 3\right) \frac{d}{d x} 5 x \]
productApply product rule✓ Proved
\[ = 125 x^{2} + 5 x \frac{d}{d x} \left(25 x^{2} - 15 x + 3\right) - 75 x + 15 \]
constant-multipleDerivative of 5*x is 5✓ Proved
\[ = 125 x^{2} + 5 x \left(50 x - 15\right) - 75 x + 15 \]
derivativeDerivative of polynomial 25*x**2 - 15*x + 3✓ Proved
\[ = 375 x^{2} - 150 x + 15 \]
algebrasimplifyMultiply out 5 and 5*x Combine like terms✓ 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 \( \displaystyle f(x) = 4 x \left(- x^{2} - 3 x - 3\right) \).
\[ \frac{d}{d x} 4 x \left(- x^{2} - 3 x - 3\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 4 \frac{d}{d x} x \left(- x^{2} - 3 x - 3\right) \]
constant-multiplePull out the constant factor 4.✓ Proved
\[ = 4 x \frac{d}{d x} \left(- x^{2} - 3 x - 3\right) + 4 \left(- x^{2} - 3 x - 3\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = - 4 x^{2} + 4 x \frac{d}{d x} \left(- x^{2} - 3 x - 3\right) - 12 x - 12 \]
derivativeDifferentiate the first part of the product.✓ Proved
\[ = - 4 x^{2} + 4 x \frac{d}{d x} \left(-3\right) + 4 x \frac{d}{d x} \left(- 3 x\right) + 4 x \frac{d}{d x} \left(- x^{2}\right) - 12 x - 12 \]
sumDifferentiate the second part of the product using the sum rule.✓ Proved
\[ = - 12 x^{2} - 24 x - 12 \]
derivativealgebraalgebrasimplifyDifferentiate each term in the sum. Simplify the expression inside the parentheses. Combine like terms. Distribute the 4 and finalize the result.✓ Proved
Answer \( - 12 \left(x + 1\right)^{2} \)
Top: the graph of f = 4x·(-x² - 3x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -4x² + 4x·(-2x - 3) - 12x - 12.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} 4 x \left(- x^{2} - 3 x - 3\right) \]
derivative✓ Proved
\[ = 4 \frac{d}{d x} x \left(- x^{2} - 3 x - 3\right) \]
constant-multiple✓ Proved
\[ = 4 x \frac{d}{d x} \left(- x^{2} - 3 x - 3\right) + 4 \left(- x^{2} - 3 x - 3\right) \frac{d}{d x} x \]
product✓ Proved
\[ = - 4 x^{2} + 4 x \left(- 2 x - 3\right) - 12 x - 12 \]
derivativesimplify✓ Proved
\[ = - 12 x^{2} - 24 x - 12 \]
algebraalgebraconstant-multiplealgebra✓ Proved
\[ = - 12 \left(x + 1\right)^{2} \]
algebra✓ 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 \( \displaystyle f(x) = x \left(25 x^{2} + 15 x + 3\right) \).
\[ \frac{d}{d x} x \left(25 x^{2} + 15 x + 3\right) \]
Start with the derivative of the function.✓ Proved
\[ = x \frac{d}{d x} \left(25 x^{2} + 15 x + 3\right) + \left(25 x^{2} + 15 x + 3\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = 25 x^{2} + x \left(\frac{d}{d x} 3 + \frac{d}{d x} 15 x + \frac{d}{d x} 25 x^{2}\right) + 15 x + 3 \]
derivativeDifferentiate the first part of the product.✓ Proved
\[ = 25 x^{2} + x \left(\frac{d}{d x} 25 x^{2} + 15\right) + 15 x + 3 \]
derivativeDifferentiate the terms in the second part.✓ Proved
\[ = 25 x^{2} + x \left(50 x + 15\right) + 15 x + 3 \]
derivativeconstant-multipleDifferentiate the power term. Simplify the first term.✓ Proved
\[ = 75 x^{2} + 30 x + 3 \]
algebrasimplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( 3 \left(5 x + 1\right)^{2} \)
Top: the graph of f = x·(25x² + 15x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 25x² + x·(50x + 15) + 15x + 3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} x \left(25 x^{2} + 15 x + 3\right) \]
derivative✓ Proved
\[ = 25 x^{2} + x \frac{d}{d x} \left(25 x^{2} + 15 x + 3\right) + 15 x + 3 \]
productApply product rule✓ Proved
\[ = 25 x^{2} + x \left(50 x + 15\right) + 15 x + 3 \]
derivativeDifferentiate the second factor✓ Proved
\[ = 75 x^{2} + 30 x + 3 \]
simplifysimplifyDistribute x over the sum Combine like terms✓ 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 \( \displaystyle f(x) = 3 x \left(- 3 x^{2} + 3 x - 1\right) \).
\[ \frac{d}{d x} 3 x \left(- 3 x^{2} + 3 x - 1\right) \]
Start with the derivative of the function.✓ Proved
\[ = 3 \frac{d}{d x} x \left(- 3 x^{2} + 3 x - 1\right) \]
constant-multiplePull out the constant factor 3.✓ Proved
\[ = 3 x \frac{d}{d x} \left(- 3 x^{2} + 3 x - 1\right) + 3 \left(- 3 x^{2} + 3 x - 1\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = - 9 x^{2} + 3 x \frac{d}{d x} \left(- 3 x^{2} + 3 x - 1\right) + 9 x - 3 \]
derivativeDifferentiate the first part of the product.✓ Proved
\[ = - 9 x^{2} + 3 x \left(3 - 6 x\right) + 9 x - 3 \]
derivativeDifferentiate the second part of the product.✓ Proved
\[ = - 27 x^{2} + 18 x - 3 \]
algebrasimplifyalgebraDistribute x into the derivative term. Combine like terms inside the parentheses. Distribute the 3 to get the final result.✓ Proved
Answer \( - 27 x^{2} + 18 x - 3 \)
Top: the graph of f = 3x·(-3x² + 3x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -9x² + 3x·(3 - 6x) + 9x - 3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} 3 x \left(- 3 x^{2} + 3 x - 1\right) \]
derivative✓ Proved
\[ = - 9 x^{2} + 3 x \left(3 - 6 x\right) + 9 x - 3 \]
productApply product rule to 3*x and the quadratic factor.✓ Proved
\[ = - 27 x^{2} + 18 x - 3 \]
simplifysimplifyDistribute the constants. Combine like terms.✓ 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 \( \displaystyle f(x) = x \left(- x^{2} + 9 x - 27\right) \).
\[ \frac{d}{d x} x \left(- x^{2} + 9 x - 27\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = x \frac{d}{d x} \left(- x^{2} + 9 x - 27\right) + \left(- x^{2} + 9 x - 27\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = - x^{2} + x \frac{d}{d x} \left(- x^{2} + 9 x - 27\right) + 9 x - 27 \]
derivativeconstant-multipleDifferentiate the first part of the product. The derivative of x is 1.✓ Proved
\[ = - x^{2} + x \left(9 - 2 x\right) + 9 x - 27 \]
derivativeDifferentiate the second part of the sum.✓ Proved
\[ = - 3 x^{2} + 18 x - 27 \]
algebrasimplifyDistribute x into the parentheses. Combine like terms.✓ Proved
Answer \( - 3 \left(x - 3\right)^{2} \)
Top: the graph of f = x·(-x² + 9x - 27). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -x² + x·(9 - 2x) + 9x - 27.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} x \left(- x^{2} + 9 x - 27\right) \]
derivative✓ Proved
\[ = x \frac{d}{d x} \left(- x^{2} + 9 x - 27\right) + \left(- x^{2} + 9 x - 27\right) \frac{d}{d x} x \]
productApply product rule✓ Proved
\[ = - x^{2} + x \frac{d}{d x} \left(- x^{2} + 9 x - 27\right) + 9 x - 27 \]
constant-multipleDerivative of x is 1✓ Proved
\[ = - x^{2} + x \left(9 - 2 x\right) + 9 x - 27 \]
derivativeDifferentiate -x**2 + 9*x - 27✓ Proved
\[ = - 3 x^{2} + 18 x - 27 \]
algebrasimplifyDistribute x Combine like terms✓ 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 \( \displaystyle f(x) = x \left(- x^{2} - 6 x - 12\right) \).
\[ \frac{d}{d x} x \left(- x^{2} - 6 x - 12\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = x \frac{d}{d x} \left(- x^{2} - 6 x - 12\right) + \left(- x^{2} - 6 x - 12\right) \frac{d}{d x} x \]
productApply the product rule.✓ Proved
\[ = - x^{2} + x \frac{d}{d x} \left(- x^{2} - 6 x - 12\right) - 6 x - 12 \]
derivativeconstant-multipleDifferentiate the first part of the product. Simplify the first term.✓ Proved
\[ = - x^{2} + x \left(- 2 x - 6\right) - 6 x - 12 \]
derivativeDifferentiate the second part of the product.✓ Proved
\[ = - 3 x^{2} - 12 x - 12 \]
constant-multiplesimplifyDistribute x into the parenthesis. Combine like terms.✓ Proved
Answer \( - 3 \left(x + 2\right)^{2} \)
Top: the graph of f = x·(-x² - 6x - 12). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -x² + x·(-2x - 6) - 6x - 12.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} x \left(- x^{2} - 6 x - 12\right) \]
derivative✓ Proved
\[ = x \frac{d}{d x} \left(- x^{2} - 6 x - 12\right) + \left(- x^{2} - 6 x - 12\right) \frac{d}{d x} x \]
productApply product rule✓ Proved
\[ = - x^{2} + x \left(- 2 x - 6\right) - 6 x - 12 \]
derivativeDifferentiate each factor✓ Proved
\[ = - 3 x^{2} - 12 x - 12 \]
algebrasimplifyMultiply x by the derivative of the second factor Combine like terms✓ 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
powerDifferentiate the first term using the power rule.✓ Proved
\[ = 3 x^{2} - 6 x + \frac{d}{d x} 3 x \]
powerDifferentiate the second term using the power rule.✓ Proved
\[ = 3 x^{2} - 6 x + 3 \]
powerDifferentiate the third term using the power rule.✓ Proved
Answer \( 3 \left(x - 1\right)^{2} \)
Top: the graph of f = x·(x² - 3x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x² + x·(2x - 3) - 3x + 3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} x \left(x^{2} - 3 x + 3\right) \]
simplifysimplifyExpand the second term Combine like terms✓ Proved
\[ = 3 \left(x - 1\right)^{2} \]
simplifyFactor the quadratic✓ 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
constantrewritePull out the constant factor 1/5. Rewrite the square root as a power.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
chainalgebraApply the chain rule. Multiply the constants.✓ Proved
\[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(5 x + 1\right)^{2}}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
sumDifferentiate the sum inside the parenthesis.✓ Proved
\[ = \frac{\frac{d}{d x} \left(5 x + 1\right)^{2}}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
constantThe derivative of the constant 1 is 0.✓ Proved
\[ = \frac{\left(10 x + 2\right) \frac{d}{d x} \left(5 x + 1\right)}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
chainApply the chain rule to the power term.✓ Proved
\[ = \frac{50 x + 10}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
derivativealgebraalgebraDifferentiate the inner linear term. Multiply the constants inside the parenthesis. Combine the terms into a single fraction.✓ Proved
\[ = \frac{5 x + 1}{\sqrt{\left(5 x + 1\right)^{2} + 1}} \]
simplifySimplify the fraction by canceling the 10.✓ Proved
Answer \( \frac{5 x + 1}{\sqrt{\left(5 x + 1\right)^{2} + 1}} \)
Top: the graph of f = sqrt((5x + 1)² + 1)/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (25x + 5)/(5sqrt((5x + 1)² + 1)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\sqrt{\left(5 x + 1\right)^{2} + 1}}{5} \]
constant-multiplePull out the constant 1/5✓ Proved
\[ = \frac{\frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
powerDifferentiate the power✓ Proved
\[ = \frac{50 x + 10}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
chainDifferentiate the inner (5*x+1)**2✓ Proved
\[ = \frac{5 x + 1}{\sqrt{\left(5 x + 1\right)^{2} + 1}} \]
simplifyrewriteSimplify constants Rewrite a**(-1/2) as 1/sqrt(a)✓ 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-multiplerewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(\left(4 x - 1\right)^{2} + 1\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]
chainalgebraApply the chain rule. Simplify the constant multiplication.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(4 x - 1\right)^{2}}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]
sumApply the sum rule to the derivative.✓ Proved
\[ = \frac{5 \left(8 x - 2\right) \frac{d}{d x} \left(4 x - 1\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]
chainApply the chain rule to the squared term.✓ Proved
\[ = \frac{5 \left(32 x - 8\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]
derivativealgebraDifferentiate the inner linear function. Multiply the constants in the derivative term.✓ Proved
\[ = \frac{20 x - 5}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \]
algebrasimplifySimplify the final expression by canceling 8/8. Rewrite with a positive exponent in the denominator.✓ Proved
Answer \( \frac{5 \left(4 x - 1\right)}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \)
Top: the graph of f = 5sqrt((4x - 1)² + 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5(16x - 4)/(4sqrt((4x - 1)² + 1)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ = \frac{5 \frac{d}{d x} \left(\left(4 x - 1\right)^{2} + 1\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]
powerconstant-multipleDifferentiate the outer power Combine constants✓ Proved
\[ = \frac{5 \left(8 x - 2\right) \frac{d}{d x} \left(4 x - 1\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]
productDifferentiate the sum inside✓ Proved
\[ = \frac{5 \left(32 x - 8\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]
derivativeDerivative of 4*x - 1 is 4✓ Proved
\[ = \frac{5 \left(4 x - 1\right)}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \]
constant-multiplesimplifyMultiply 2 and 4 Cancel 8 with 8✓ Proved
\[ = \frac{20 x - 5}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \]
rewriteRewrite power -1/2 as 1/sqrt✓ 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
chainApply the chain rule and power rule to the inner function.✓ Proved
\[ = \frac{2 x}{\sqrt{4 x^{2} + 1}} \]
powersimplifyalgebrarewriteCompute the derivative of the inner polynomial 4x^2 + 1. Multiply the constants 1/2 and 1/2. Simplify the product of 1/4 and 8x to 2x. Rewrite the negative exponent as a square root in the 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
constant-multiplerewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(\left(3 x + 2\right)^{2} + 1\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
chainApply the chain rule.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(3 x + 2\right)^{2}}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
constant-multipleSimplify the constant coefficient.✓ Proved
\[ = \frac{5 \left(6 x + 4\right) \frac{d}{d x} \left(3 x + 2\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
chainApply the chain rule to the inner squared term.✓ Proved
\[ = \frac{5 \left(18 x + 12\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
derivativealgebraDifferentiate the innermost linear term. Multiply the constants in the derivative term.✓ Proved
\[ = \frac{15 x + 10}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \]
simplifyrewriteSimplify the final expression by canceling the 6 and 6/3. Rewrite the negative exponent as a denominator.✓ Proved
Answer \( \frac{5 \left(3 x + 2\right)}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \)
Top: the graph of f = 5sqrt((3x + 2)² + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5(9x + 6)/(3sqrt((3x + 2)² + 1)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ = \frac{5 \frac{d}{d x} \left(\left(3 x + 2\right)^{2} + 1\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
chain✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(3 x + 2\right)^{2}}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
sum✓ Proved
\[ = \frac{5 \left(3 x + 2\right) \frac{d}{d x} \left(3 x + 2\right)}{3 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
product✓ Proved
\[ = \frac{5 \left(3 x + 2\right)}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \]
constant✓ Proved
\[ = \frac{15 x + 10}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \]
simplify✓ Proved
\[ = \frac{15 x + 10}{\sqrt{9 x^{2} + 12 x + 5}} \]
rewrite✓ 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