Quotient rule practice problems
The derivative of a quotient: (f/g)′ = (f′g − fg′)/g². 53 problems with worked solutions; in 53 of them every equation is proved by a computer algebra system.
Differentiate \( \displaystyle f(x) = \frac{10 x + 3}{5 x - 4} \).
Differentiate \( \displaystyle f(x) = \frac{2 x + 1}{2 x - 1} \).
Differentiate \( \displaystyle f(x) = \frac{x + 1}{x - 1} \).
Differentiate \( \displaystyle f(x) = \frac{8 x + 7}{4 x - 2} \).
Differentiate \( \displaystyle f(x) = \frac{5 \operatorname{atan}{\left(4 x \right)}}{4} \).
Differentiate \( \displaystyle f(x) = \frac{\operatorname{atan}{\left(2 x \right)}}{2} \).
Differentiate \( \displaystyle f(x) = - \operatorname{atan}{\left(x \right)} \).
Differentiate \( \displaystyle f(x) = \frac{5 x}{5 x + 1} \).
Differentiate \( \displaystyle f(x) = \frac{2 x + 5}{x - 3} \).
Differentiate \( \displaystyle f(x) = 5 \operatorname{atan}{\left(x + 1 \right)} \).
Differentiate \( \displaystyle f(x) = \frac{3 \operatorname{atan}{\left(2 x - 1 \right)}}{2} \).
Differentiate \( \displaystyle f(x) = \frac{5 \operatorname{atan}{\left(2 x + 2 \right)}}{2} \).
Differentiate \( \displaystyle f(x) = 5 \operatorname{atan}{\left(x - 3 \right)} \).
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(2 x + 1 \right)} \).
Differentiate \( \displaystyle f(x) = \frac{\operatorname{atan}{\left(5 x \right)}}{5} \).
Differentiate \( \displaystyle f(x) = \frac{x + 2}{x} \).
Differentiate \( \displaystyle f(x) = \frac{\operatorname{atan}{\left(3 x - 1 \right)}}{3} \).
Differentiate \( \displaystyle f(x) = 3 \operatorname{atan}{\left(x + 1 \right)} \).
Differentiate \( \displaystyle f(x) = - \frac{\operatorname{atan}{\left(3 x - 3 \right)}}{3} \).
Differentiate \( \displaystyle f(x) = \frac{2 \operatorname{atan}{\left(5 x \right)}}{5} \).
Differentiate \( \displaystyle f(x) = \frac{5 \operatorname{atan}{\left(3 x + 1 \right)}}{3} \).
Differentiate \( \displaystyle f(x) = 2 \operatorname{atan}{\left(x \right)} \).
Differentiate \( \displaystyle f(x) = \frac{3 \operatorname{atan}{\left(4 x \right)}}{4} \).
Differentiate \( \displaystyle f(x) = \frac{3 \operatorname{atan}{\left(5 x - 1 \right)}}{5} \).
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x + 1 \right)}}{2 x + 1} \).
Differentiate \( \displaystyle f(x) = \frac{e^{x}}{x} \).
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(4 x + 1 \right)}}{4 x + 2} \).
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{x + 1} \).
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x - 1 \right)}}{2 x - 1} \).
Differentiate \( \displaystyle f(x) = \frac{x^{2} - 1}{x^{2} + 1} \).
Differentiate \( \displaystyle f(x) = x e^{- x} \).
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(4 x \right)}}{4 x} \).
Differentiate \( \displaystyle f(x) = \frac{e^{2 x}}{2 x} \).
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(x \right)}}{x} \).
Differentiate \( \displaystyle f(x) = \frac{\tan{\left(x \right)}}{x} \).
Differentiate \( \displaystyle f(x) = \left(x - 3\right) e^{3 - x} \).
Differentiate \( \displaystyle f(x) = \frac{\sqrt{4 x + 1}}{4 x + 2} \).
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3} \).
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(4 x + 1 \right)} \right)}}{4} \).
Differentiate \( \displaystyle f(x) = \frac{e^{x}}{x + 1} \).
Differentiate \( \displaystyle f(x) = \frac{4 x^{2} - 1}{4 x^{2} + 1} \).
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(x + 2 \right)}}{x + 2} \).
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x + 2 \right)}}{2 x + 3} \).
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(4 x \right)}}{4 x} \).
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x - 1 \right)}}{2 x - 1} \).
Differentiate \( \displaystyle f(x) = \frac{\sqrt{x - 1}}{x} \).
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x + 2 \right)}}{2 x + 2} \).
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} \sqrt{x}}{2 x + 1} \).
Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(3 x - 3 \right)} \right)} \).
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x - 3 \right)}}{2 x - 3} \).
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\ln{\left(2 x - 3 \right)} \right)}}{2} \).
Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(x - 1 \right)} \right)} \).
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(5 x - 1 \right)} \right)}}{5} \).