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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 at $a$.

## Notation

There are three types of notation typically used:

• Leibniz's notation:
$\frac{d}{dx}(f(x)) = \frac{df}{dx}$
$\frac{d^n}{dx^n} (f(x)) = \frac{d^n f}{dx^n}$
• Lagrange's notation:
$\frac{d}{dx} (f(x))= f'(x)$
$\frac{d^n}{dx^n} (f(x)) = f^n (x)$
• Newton's notation:
$\frac{d}{dt} x(t) = \dot{x}(t)$

Newton's notation is typically used in areas of physics, especially in differential equations. However, the notation becomes unwieldy with higher order derivatives.

## Properties

The derivative operator is linear, that is the derivative of the sum is the sum of the derivatives:

$\frac{d}{dx} (f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$

For example, using the derivative of a polynomial term:

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

The function $f(x)=5x^3+2x^2+4x+6$ can be differentiated as follows:

\begin{align} \frac{d}{dx} f(x)&=\frac{d}{dx}(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}

Applying the derivative operator multiple times gives higher order derivatives. For example, a second order derivative can be found as follows

$\frac{d}{dx}\left(\frac{d}{dx} f(x)\right) = \frac{d}{dx} \left(\frac{df}{dx}\right) = \frac{d^2 f}{dx^2}$

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 $(a,f(a))$.

Differentiation is the inverse of Integration.