Calculus/Derivatives

Derivative of a function $f$ at a number $a$ Notation

We denote the derivative of a function $f$ at a number $a$ as $f'(a)\,\!$ .

Definition

The derivative of a function $f$ at a number $a$ a is given by the following limit (if it exists):

$f'(a)=\lim _{h\rightarrow 0}{\frac {f(a+h)-f(a)}{h}}$ An analagous equation can be defined by letting $x=(a+h)$ . Then $h=(x-a)$ , which shows that when $x$ approaches $a$ , $h$ approaches $0$ :

$f'(a)=\lim _{x\rightarrow a}{\frac {f(x)-f(a)}{x-a}}$ Interpretations

As the slope of a tangent line

Given a function $y=f(x)\,\!$ , the derivative $f'(a)\,\!$ can be understood as the slope of the tangent line to $f(x)$ at $x=a$ :

Example

Find the equation of the tangent line to $y=x^{2}$ at $x=1$ .

Solution

To find the slope of the tangent, we let $y=f(x)$ and use our first definition:

$f'(a)=\lim _{h\rightarrow 0}{\frac {f(a+h)-f(a)}{h}}\Rightarrow \lim _{h\rightarrow 0}{\frac {\color {Blue}(a+h)^{2}-(a)^{2}}{h}}\Rightarrow \lim _{h\rightarrow 0}{\frac {\color {Blue}a^{2}+2ah+h^{2}-a^{2}}{h}}\Rightarrow \lim _{h\rightarrow 0}{\frac {\color {Blue}h(2a+h)}{h}}\Rightarrow \lim _{h\rightarrow 0}{\color {Blue}(2a+h)}$ It can be seen that as $h$ approaches $0$ , we are left with $f'(a)={\color {Blue}2a}\,\!$ . If we plug in $1$ for $a$ :

$f'({\color {Red}1})=2({\color {Red}1})\Rightarrow {\color {Red}2}$ So our preliminary equation for the tangent line is $y={\color {Red}2}x+b$ . By plugging in our tangent point $(1,1)$ to find $b$ , we can arrive at our final equation:

${\color {Red}1}=2({\color {Red}1})+b\Rightarrow -1=b$ So our final equation is $y=2x-1\,\!$ .

As a rate of change

The derivative of a function $f(x)$ at a number $a$ can be understood as the instantaneous rate of change of $f(x)$ when $x=a$ .

The derivative as a function

So far we have only examined the derivative of a function $f$ at a certain number $a$ . If we move from the constant $a$ to the variable $x$ , we can calculate the derivative of the function as a whole, and come up with an equation that represents the derivative of the function $f$ at any arbitrary $x$ value. For clarification, the derivative of $f$ at $a$ is a number, whereas the derivative of $f$ is a function.

Notation

Likewise to the derivative of $f$ at $a$ , the derivative of the function $f(x)$ is denoted $f'(x)\,\!$ .

Definition

The derivative of the function $f$ is defined by the following limit:

$f'(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}$ Also,

$f'(x)=\lim _{h\rightarrow x}{\frac {f(x)-f(h)}{h-x}}$ or

$f'(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x-h)}{2h}}$ 