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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.