Differentiation Rules

Which of the following is CORRECT?

• $\frac{d}{dx}\left(-5\right)=-5$

• $D_x\left(4\right)=-4$

• $\frac{d}{dx}\left(\frac{1}{2}x\right)=1$

• $D_x\left(\sqrt{2}\right)=0$

 $\frac{d}{dx}\left(\frac{2}{3}x+1\right)$  

Find the derivative.

•  $1$  

•  $\frac{1}{2}$  

•  $\frac{2}{3}$  

•  $\frac{5}{3}$  

 $f\left(x\right)=4x^3-3x^2+\sqrt{2}x$  , find its derivative.

•  $f'\left(x\right)=12-6x$  

•  $f'\left(x\right)=12x^2-6x+\sqrt{2}$  

•  $f\left(x\right)=12x^2-6x+\sqrt{2}$  

•  $f'\left(x\right)=3x^4-2x^3+x$  

 $y=\frac{2}{\sqrt{x}}$  

What is the derivative of the given function?

•  $y\ '=-\frac{1}{x\sqrt{x}}$  

•  $\frac{dy}{dx}=\frac{1}{\sqrt{x^3}}$  

•  $y\ '=\frac{1}{x\sqrt{x}}$  

•  $\frac{dy}{dx}=-\frac{1}{2\sqrt{x}}$  

 $D_x\left[\left(x\right)\left(x^3+4x\right)\right]$  

Find the derivative.

•  $x^4+4x^2$  

•  $4x^3+8x$  

•  $12x^2+8$  

•  $24x$  

 $\frac{d}{dx}\left(e^{2x^{ }}\right)$  
Find the derivative.

•  $e^{2x}$  

•  $2e^{2x}x$  

•  $2e^{2x}$  

•  $2x$  

 $D_x\left(3^{3x}\right)$  
Find its derivative.

•  $3^{\left(3x+1\right)}$  

•  $3^{\left(3x+1\right)}\ln\left(3\right)$  

•  $3^{3x}\ln\left(3\right)$  

•  $3^{3x}\ln\left(9\right)$  

 $y=\ln\left(4x+4\right)$  
Find y'.

•  $y'=\frac{1}{x+1}$  

•  $y'=\frac{4}{4x+1}$  

•  $y'=\frac{1}{\left(x+1\right)\ln\left(4\right)}$  

•  $y'=\frac{1}{x}$  

 $y=\log_5\left(5x\right)$  
Find the derivative.

•  $y'=\frac{1}{x\ln\left(5x\right)}$  

•  $y'=\frac{1}{\ln\left(5\right)\cdot x}$  

•  $y'=\frac{5}{5x\cdot\ln\left(x\right)}$  

•  $y'=\ln\left(5x\right)$  

 $D_x\left(\sin\left(4x^3\right)\right)$  is equal to what?

•  $\sin\left(4x^3\right)\left(12x\right)$  

•  $12x^2\cos\left(4x^3\right)$  

•  $\cos\left(48x^5\right)$  

•  $12x\cos\left(4x^3\right)$