I've noticed some of my students have trouble with identifying when to use the product rule or when to use the chain rule. Here are some practise questions.
In each of the expressions below state if the chain rule or the product rule is required to find the derivative of it. Also, state what the $u$ and the $v$ are.
Q1) $x \sin x$
Q2) $\sqrt{\sin x}$
Q3) $\sqrt{x} e^x$
Q4) $e^{\sin x}$
Q5) $e^x \ln x$
Q6) $\sin \ln x$
Q7) $\sin x \ln x$
Q8) $\sin x^2$
Q9) $\tan \sin x$
Q10) $\left(e^x + e^{-x}\right)^3$
Q11) $x e^x$
Q12) $e^{x^2}$
Q13) $\frac{1}{x} e^x$
Q14) $\ln \frac{1}{x}$
Q15) $x^3 \ln x$
Q16) $\left(\ln x\right)^3$
Q17) $\ln \ln x$
Q18) $\sqrt{1+x^3}$
Q19) $\sin^2 x$
Q20) $\left(\ln x \right)^2$
A1) $x \sin x$ Product Rule $u= x$ and $v= \sin x$
A2) $\sqrt{\sin x}$ Chain Rule $u= \sqrt{v}$ and $v= \sin x$
A3) $\sqrt{x} e^x$ Product Rule $u= \sqrt{x}$ and $v= e^x$
A4) $e^{\sin x}$ Chain Rule $u= e^v$ and $v= \sin x$
A5) $e^x \ln x$ Product Rule $u= e^x$ and $v= \ln x$
A6) $\sin \ln x$ Chain Rule $u= \sin v$ and $v= \ln x$
A7) $\sin x \ln x$ Product Rule $u= \sin x$ and $v= \ln x$
A8) $\sin x^2$ Chain Rule $u= \sin v$ and $v= x^2$
A9) $\tan \sin x$ Chain Rule $u= \tan v$ and $v= \sin x$
A10) $\left(e^x + e^{-x}\right)^3$ Chain Rule $u=v^2 $ and $v= e^x + e^{-x}$
A11) $x e^x$ Product Rule $u= x$ and $v= e^x$
A12) $e^{x^2}$ Chain Rule $u= e^v$ and $v=x^2 $
A13) $\frac{1}{x} e^x$ Product Rule $u= \frac{1}{x}$ and $v= e^x$
A14) $\ln \frac{1}{x}$ Chain Rule $u= \ln v$ and $v= \frac{1}{x}$
A15) $x^3 \ln x$ Product Rule $u= x^3$ and $v= \ln x$
A16) $\left(\ln x\right)^3$ Chain Rule $u= v^3 $ and $v= \ln x $
A17) $\ln \ln x$ Chain Rule $u= \ln v$ and $v= \ln x$
A18) $\sqrt{1+x^3}$ Chain Rule $u= \sqrt{v}$ and $v= 1+x^3$
A19) $\sin^2 x$ Product Rule $u= \sin x$ and $v= \sin x$ or Chain Rule $u=v^2$ and $v=\sin x$
A20) $\left(\ln x \right)^2$ Chain Rule $u= v^2$ and $v= \ln x$ or Product Rule $u=\ln x$ and $v=\ln x$
Product or Chain Rule
Posted
Author Stephen Easley-Walsh