I.2. Common derivatives (Review of Derivatives)

Find the derivative of  $f\left(x\right)=c$  , where  $c$   is a constant.

•  $f'\left(x\right)=0$  

•  $f'\left(x\right)=c$  

•  $f'\left(x\right)=1$  

•  $f'\left(x\right)=cx$  

Find the derivative of $f\left(x\right)=x^n$   .

•  $f'\left(x\right)=nx$  

•  $f'\left(x\right)=nx^{n+1}$  

•  $f'\left(x\right)=nx^{n-1}$  

•  $f'\left(x\right)=x^{n-1}$  

What is the derivative of  $c\cdot f\left(x\right)$  , where  $c$  is a constant?

•  $0$  

•  $c$  

•  $c\cdot f'\left(x\right)$  

•  $f'\left(x\right)$  

What is the derivative of the function  $\left(f+g\right)\left(x\right)?$  

•  $f'\left(x\right)+g'\left(x\right)$  

•  $f'\left(x\right)-g'\left(x\right)$  

•  $f'\left(x\right)\cdot g\left(x\right)+f\left(x\right)\cdot g'\left(x\right)$  

•  $f'\left(x\right)\cdot g\left(x\right)-f\left(x\right)\cdot g'\left(x\right)$  

The product rule: What is the derivative of $\left(f\cdot g\right)\left(x\right)?$  

•  $f'\left(x\right)\cdot g'\left(x\right)$  

•  $f'\left(x\right)\cdot g\left(x\right)+f\left(x\right)\cdot g'\left(x\right)$  

The quotient rule: What is the derivative of $\left(\frac{f}{g}\right)\left(x\right)?$  

•  $\frac{f'\left(x\right)\cdot g\left(x\right)}{f\left(x\right)\cdot g'\left(x\right)}$  

•  $\frac{f'\left(x\right)}{g'\left(x\right)}$  

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

In the questions above, what did we assume about the functions  $f$   and $g?$  

• They are both continuous

• They are both differentiable

• They are not equal

• Neither are constant functions

What is the derivative of $f\left(x\right)=\sin\left(x\right)?$  Include the signs when answering.

• -

• +

•  $\sin\left(x\right)$  

•  $\cos\left(x\right)$  

•  $\tan\left(x\right)$  

What is the derivative of $f\left(x\right)=\cos\left(x\right)?$  Include the signs when answering.

• -

• +

•  $\sin\left(x\right)$  

•  $\cos\left(x\right)$  

•  $\sec\left(x\right)$  

What is the derivative of $f\left(x\right)=\tan\left(x\right)?$  Include the signs when answering.

• -

• +

•  $\cot\left(x\right)$  

•  $\sec\left(x\right)$ 

•  $\sec^2\left(x\right)$