Definition of a Derivative and Limits

Quote:

Originally Posted by trevor22

I at least need help getting started.

http://i43.tinypic.com/97r6hg.jpg

For the first, use $\displaystyle \begin{align*} \frac{dy}{dt} = \lim_{h \to 0}\frac{y(t + h) - y(t)}{h} \end{align*}$

For the second...

$\displaystyle \begin{align*} 3^x\lim_{h \to 0}\frac{3^h - 1}{h} &= \lim_{h \to 0}\frac{3^x\left(3^h - 1\right)}{h} \\ &= \lim_{h \to 0}\frac{3^x3^h - 3^x}{h} \\ &= \lim_{h \to 0}\frac{3^{x+h} - 3^x}{h} \end{align*}$

Is this in the form $\displaystyle \begin{align*} \lim_{h \to 0}\frac{f(x+h) - f(x)}{h} \end{align*}$ ? If so, what is $\displaystyle \begin{align*} f(x) \end{align*}$ ?