Main Concept

Given a function $f\left(x\right)$, its derivative, denoted $\frac{\mathrm{df}}{\mathrm{dx}}$ or $f\'\left(x\right)$, is a new function describing the rate of change of $f\left(x\right)$.

The value of the derivative $\frac{\mathrm{df}}{\mathrm{dx}}$ at any point $x$ is defined by the following limit, if it exists:

$\underset{h\to 0}{lim} \frac{f\left(x\+h\right)- f\left(x\right)}{h}$

Geometrically, $\frac{\mathrm{df}}{\mathrm{dx}}$ describes the slope of the tangent to the graph of  $f\left(x\right)$.

You can find an approximation to the value of the derivative by ignoring the limit:

$\frac{f\left(x\+h\right)- f\left(x\right)}{h}$

This expression is the slope of the secant from P = $\left(x\,f\left(x\right)\right)$to a nearby point such as Q = $\left(x\+h\,f\left(x\+h\right)\right)$, and the approximation improves as $h$ becomes smaller.

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