Power, Product, and Quotient Rule

Which of the following correctly states the power rule?

• $\frac{d}{dx}\left(x^n\right)=\ln\left(x\right)\cdot x^n$

• $\frac{d}{dx}\left(x^n\right)=nx^n$

• $\frac{d}{dx}\left(x^n\right)=nx^{n-1}$

• $\frac{d}{dx}\left(x^n\right)=x^{n-1}$

Which of the following correctly states the product rule?

• $\frac{d}{dx}\left(f\cdot g\right)\ =\ f'\cdot g'$

• $\frac{d}{dx}\left(f\cdot g\right)\ =\ f\cdot g'\ +g'\cdot f$

• $\frac{d}{dx}\left(f\cdot g\right)\ =\ f'\cdot g^2-f\cdot g$

• $\frac{d}{dx}\left(f\cdot g\right)\ =\ f'\cdot g\ +\ f\cdot g'$

Which of the following correctly states the quotient rule?

• $\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{\ g\cdot f'\ -\ f\cdot\ g'}{g^2}$

• $\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{\ g'\cdot f\ -\ g\cdot\ f'}{g^2}$

• $\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{\ g'\cdot f\ -\ g\cdot\ f'}{f^2}$

• $\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{\ g\cdot f'\ -\ f\cdot g'}{f^2}$

Find the derivative of

 $f\left(x\right)\ =\ e^x$  

•  $xe^x$  

•  $xe^{x-1}$  

•  $e^x$  

•  $e$  

Find the derivative of

 $f\left(x\right)\ =\ \cos\left(x\right)$  

•  $\cos x$  

•  $\sin x$  

•  $-\cos x$  

•  $-\sin x$  

Find the derivative of

 $f\left(x\right)\ =\ \ln x$  

•  $\ln$  

•  $x$  

•  $\frac{1}{x}$  

•  $\ln\left(\ln\left(x\right)\right)$  

Find the derivative of

 $f\left(x\right)\ =\ \tan x$  

•  $\sec^2x$  

•  $\csc^2x$  

•  $\cot^2x$  

•  $1$  

Find the derivative of
 $f\left(x\right)\ =\ \frac{3\sqrt{x}}{x}$ .

•  $\frac{3}{\sqrt{x}}$  

•  $3\sqrt{x}$  

•  $\frac{3}{2}x^{-\frac{3}{2}}$  

•  $-\ \frac{3}{2}x^{-\frac{3}{2}}$  

Find the derivative of
 $f\left(x\right)\ =\ \cos^2x.$  
(hint:  $\cos^2x\ =\ \cos x\cdot\cos x$ )

•  $-2\sin x\cos x$  

•  $2\sin x\cos x$  

•  $\sin^2x$  

•  $-\sin^2x$  

Find the derivative of 
 $f\left(x\right)\ =\ \ln e\cdot\cos\pi.$ 

•  $\frac{\sin\pi}{e}$  

•  $-\frac{\sin\pi}{e}$  

•  $\frac{\cos\pi}{e}-\ln e\sin\pi$  

•  $0$