Originally Posted by

**MarkFL2** c) Your first two terms are correct. For the third term, rewrite it as:

$\displaystyle \log_5(e^x)=x\log_5(e)$

Now you just have a constant times *x*.

For the 4th term, let's write:

$\displaystyle y=2^{\sqrt{x}}$

take the natural log of both sides:

$\displaystyle \ln(y)=\sqrt{x}\ln(2)$

Implicitly differentiate:

$\displaystyle \frac{1}{y}\cdot\frac{dy}{dx}=\frac{\ln(2)}{2\sqrt {x}}$

$\displaystyle \frac{dy}{dx}=y\frac{\ln(2)}{2\sqrt{x}}$

$\displaystyle \frac{dy}{dx}=2^{\sqrt{x}}\frac{\ln(2)}{2\sqrt{x}}$

d) For the first term you need to apply the product rule, and for the second term use:

$\displaystyle \frac{d}{dx}\left(\tan^{-1}(u(x)) \right)=\frac{1}{u^2+1}\cdot\frac{du}{dx}$

e) Apply the quotient rule.

f) Correct.