Chain Rule, Implicit, Logs and Exponential Derivatives

Differentiate

 $y=3^{\left(x^2-1\right)}$  

•  $x^2\left(\ln3\right)\left(3^{x^2-1}\right)$  

•  $\left(\ln3\right)\left(3^{x^2-1}\right)$  

•  $\left(2x\right)\left(3^{x^2-1}\right)$  

•  $2x\left(\ln3\right)\left(3^{x^2-1}\right)$  

Find  $\frac{\text{d}y}{\text{d}x}$   if  $y=e^{\sin\sqrt{x}}$  

•  $\left(\cos\sqrt{x}\right)e^{\sin\sqrt{x}}$  

•  $\frac{e^{\cos\sqrt{x}}}{2\sqrt{x}}$  

•  $\frac{\cos\sqrt{x}}{2\sqrt{x}}\left(e^{\sin\sqrt{x}}\right)$  

•  $\left(\sin\sqrt{x}\right)e^{\left(\sin\sqrt{x}-1\right)}$  

 $f\left(x\right)=\sqrt{5+e^{2x}}$  

Find f'(x) for

•  $\frac{1}{2\sqrt{2e^{2x}}}$  

•  $\frac{e^{2x}}{\sqrt{5+e^{2x}}}$  

•  $\frac{5+e^{2x}}{\sqrt{5+e^{2x}}}$  

•  $\frac{xe^{\left(2x-1\right)}}{\sqrt{5+e^{2x}}}$  

If $y=\tan^{-1}\left(\cos\ x\right)$  , then  $\frac{\text{d}y}{\text{d}x}$  =

•  $\left(\sec^{-1}\left(\cos\ x\right)\right)^2$  

•  $\frac{1}{\left(\cos^{-1}x\right)^2+1}$  

•  $-\left(\sec^{-1}\left(\cos x\right)\right)^2\sin x$  

•  $\frac{-\sin x}{1+\cos^2x}$  

Let $f\left(x\right)=x^4+3x-2$  , and let g(x) denote the inverse of f.  Then g'(2) is equal to:

• 1/7

• 1/35

• 7

• 35

 $g\left(x\right)=\sin^3\left(4x\right)$  

Find g'(x) if

•  $\cos^3\left(4x\right)$  

•  $3\sin^2\left(4x\right)\cos\left(4x\right)$  

•  $12\sin^2\left(4x\right)\cos\left(4x\right)$  

•  $4\cos^3\left(4x\right)$  

For what values of x does $y^2-2x^3-12x^2=0$  have horizontal tangent lines?

• x = 0 only

• x = 0 and x = - 4

• x = - 4 only

• x = - 4, x = 0, and x = 4

If $f\left(x\right)=\left(x+1\right)\left(x^2-2\right)^3$  then f'(x)=

•  $6x\left(x^2-2\right)^2$  

•  $6x\left(x+1\right)\left(x^2-2\right)^2$  

•  $3\left(x+1\right)\left(x^2-2\right)^2$  

•  $\left(x^2-2\right)^2\left(7x^2+6x-2\right)$  

Find the second derivative give $g\left(x\right)=\sin\ x^2$  

•  $2\left(\cos\ x^2-x\sin x^2\right)$  

•  $2\left(\cos x^2-2x^2\sin x^2\right)$  

•  $2x\cos x^2$  

•  $-4x\sin x^2$  

If $h\left(x\right)=\ln\left(2-3x\right)^5$  , find h'(0)

• 5/2

• -2/3

• -5/3

• -15/2