# Derivative

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A derivative of a function is a second function showing the rate of change of the dependent variable compared to the independent variable. It can be thought of as a graph of the slope of the function from which it is derived. The process of finding a derivative is called differentiation.

Definition
Let $D\subseteq\R$ and $f:D\to\R$ be a function. Then the derivative of $f$ is a function $f':D'\to\R$ defined by:
$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$.

The derivative is undefined when this limit does not exist, that is, $f$ is not differentiable.

## Properties

A simpler way of finding the derivative is to multiply the coefficient by the exponent and subtract one from the exponent (proof). For example:

$(x^r)'=rx^{r-1}$

The function

$f(x)=5x^3+2x^2+4x+6$

can be differentiated as follows:

\begin{align}f(x)&=5x^3+2x^2+4x+6\\ f'(x)&=(3)5x^{3-1}+(2)2x^{2-1}+(1)4x^{1-1}+(0)6\\ &=15x^2+4x^1+4x^0+0\\ &=15x^2+4x+4\end{align}

In graphs, the derivative of a function $f$ at a number $a$ is equal to the slope of the tangent line of the graph of $f$ at the point $(a,f(a))$ .