# Definition of a Derivative and Limits

• December 13th 2011, 01:51 PM
trevor22
Definition of a Derivative and Limits
I at least need help getting started.

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• December 13th 2011, 03:11 PM
Prove It
Re: 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*}?