Product rule practice problems
The derivative of a product: (fg)′ = f′g + fg′. 61 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.
Differentiate \( \displaystyle f(x) = 2 x \left(x + 2\right) \).
Differentiate \( \displaystyle f(x) = 2 x \left(1 - x\right) \).
Differentiate \( \displaystyle f(x) = x \left(3 x + 2\right) \).
Differentiate \( \displaystyle f(x) = 4 x \left(x + 1\right) \).
Differentiate \( \displaystyle f(x) = x \left(5 x + 2\right) \).
Differentiate \( \displaystyle f(x) = 3 x \left(x - 2\right) \).
Differentiate \( \displaystyle f(x) = 10 x \left(x - 3\right) \).
Differentiate \( \displaystyle f(x) = 2 x \left(3 x + 2\right) \).
Differentiate \( \displaystyle f(x) = 6 x \left(x - 2\right) \).
Differentiate \( \displaystyle f(x) = 6 x \left(x - 1\right) \).
Differentiate \( \displaystyle f(x) = 2 x \left(x + 4\right) \).
Differentiate \( \displaystyle f(x) = x \left(x - 6\right) \).
Differentiate \( \displaystyle f(x) = \frac{x}{x + 1} \).
Differentiate \( \displaystyle f(x) = \sin{\left(x \right)} \cos{\left(x \right)} \).
Differentiate \( \displaystyle f(x) = \frac{5 x + 1}{5 x - 1} \).
Differentiate \( \displaystyle f(x) = \frac{2 x}{2 x + 1} \).
Differentiate \( \displaystyle f(x) = \frac{4 x + 3}{2 x - 4} \).
Differentiate \( \displaystyle f(x) = \frac{x + 1}{\sqrt{x}} \).
Differentiate \( \displaystyle f(x) = \left(x - \frac{1}{3}\right) e^{3 x} \).
Differentiate \( \displaystyle f(x) = 3 x e^{3 x + 1} \).
Differentiate \( \displaystyle f(x) = \left(x + 1\right) e^{x + 2} \).
Differentiate \( \displaystyle f(x) = \left(x - 1\right) e^{4 x - 3} \).
Differentiate \( \displaystyle f(x) = 5 \left(x - 4\right) e^{x - 3} \).
Differentiate \( \displaystyle f(x) = \left(x + \frac{1}{2}\right) e^{2 x + 2} \).
Differentiate \( \displaystyle f(x) = \left(3 x + 2\right) \tan{\left(3 x + 2 \right)} \).
Differentiate \( \displaystyle f(x) = \left(5 x + 2\right) \sin{\left(5 x + 2 \right)} \).
Differentiate \( \displaystyle f(x) = 5 x \sqrt{5 x + 1} \).
Differentiate \( \displaystyle f(x) = \left(3 x - \frac{3}{4}\right) e^{4 x} \).
Differentiate \( \displaystyle f(x) = \left(\frac{1}{3} - x\right) e^{3 x} \).
Differentiate \( \displaystyle f(x) = 5 x \sin{\left(5 x \right)} \).
Differentiate \( \displaystyle f(x) = \left(5 x - \frac{5}{3}\right) e^{3 x} \).
Differentiate \( \displaystyle f(x) = \left(\frac{1}{4} - x\right) e^{4 x} \).
Differentiate \( \displaystyle f(x) = \left(x + \frac{1}{3}\right) e^{3 x + 2} \).
Differentiate \( \displaystyle f(x) = \left(- x - \frac{1}{5}\right) e^{5 x + 2} \).
Differentiate \( \displaystyle f(x) = 3 \left(x - 4\right) e^{x - 3} \).
Differentiate \( \displaystyle f(x) = \left(2 - x\right) e^{2 x - 3} \).
Differentiate \( \displaystyle f(x) = \left(5 x - \frac{5}{2}\right) e^{2 x} \).
Differentiate \( \displaystyle f(x) = 3 \left(x - 2\right) e^{2 x - 3} \).
Differentiate \( \displaystyle f(x) = 5 x e^{5 x + 1} \).
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(x \right)}}{x + 1} \).
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \).
Differentiate \( \displaystyle f(x) = x^{2} \left(2 \ln{\left(2 x \right)} - 1\right) \).
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \).
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \).
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \).
Differentiate \( \displaystyle f(x) = x^{3} e^{x} \).
Differentiate \( \displaystyle f(x) = x^{2} \left(3 \ln{\left(2 x \right)} - \frac{3}{2}\right) \).
Differentiate \( \displaystyle f(x) = 9 x^{2} \cos{\left(3 x \right)} \).
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \).
Differentiate \( \displaystyle f(x) = \ln{\left(x \right)} \sin{\left(x \right)} \).
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10} \).
Differentiate \( \displaystyle f(x) = \sqrt{2 x - 1} \ln{\left(2 x - 1 \right)} \).
Differentiate \( \displaystyle f(x) = 8 x^{3} e^{2 x} \).
Differentiate \( \displaystyle f(x) = \left(x + 1\right)^{3} e^{x + 1} \).
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \).
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \).
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \).
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \).
Differentiate \( \displaystyle f(x) = \frac{3 x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \).
Differentiate \( \displaystyle f(x) = x^{2} \left(1 - 2 \ln{\left(4 x \right)}\right) \).
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{2} - x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{x}{2} - \frac{\ln{\left(2 x - 1 \right)}}{4} \).