Derivative/Definition

The derivative of a function is defined as the instantaneous rate of change of the function at a certain point. For a line, this is just the slope. For more complex curves, we can find the rate of change between two points on the curve easily since we can draw a line through them.

In the image above, the average rate of change between the two points is the slope of the line that goes through them: $\frac{f(x+h)-f(x)}h$ .

We can move the second point closer to the first one to find a more accurate value of the derivative. Thus, taking the limit as $h$ goes to 0 will give us the derivative of the function at $x$ :

$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}h.$

If this limit exists, it is the derivative of $f$ at $x$ . If it does not exist, we say that $f$ is not differentiable at $x$ .

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Derivative/Definition

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