Difference Quotient

Slope of tangent at a given point x can be modeled with: $m={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Proof

This is essentially a rise/run where we try to reduce the run distance to 0 and get tangent. Get the value as h(the run distance) approaches 0.

Example

$f(x)=\sqrt{x}$ what is slope at x = 9?

Solution

${\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$ ${\lim }_{h\to 0}\frac{\sqrt{9+h}-\sqrt{9}}{h}$ ${\lim }_{h\to 0}\frac{\sqrt{9+h}-3}{h}\ast \frac{\sqrt{9+h}+3}{\sqrt{9+h}+3}$ ${\lim }_{h\to 0}\frac{9+h-9}{h(\sqrt{9+h}+3)}$ ${\lim }_{h\to 0}\frac{h}{h(\sqrt{9+h}+3)}$ ${\lim }_{h\to 0}\frac{1}{\sqrt{9+h}+3}$ $\frac{1}{\sqrt{9}+3}$ $\frac{1}{6}$