Derivative

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Sub Topics Derivative Definition Derivative Rules Derivative Table Derivative of Fraction Derivative Examples

A derivative is the instantaneous rate of change of a function. For example while finding rates of change, the instantaneous rate of change of a function f(x) when x = a is equal to the derivative of f at a. If f(x) = y and for a small change in x say $\partial$x , the function changes by $\partial$y. If the $\lim_{\partial x \rightarrow 0}$ $\frac{\partial y}{\partial x}$ exists then it is called as the derivative of the function. Derivatives have their own notation. The derivative of a function f at a number a is denoted by f'(a) and is given by:   $f'(a) = $$\lim_{h \rightarrow 0}$$\frac{f(a+h) - f(a)}{h}$ = $\lim_{x \rightarrow a}$$\frac{f(x) - f(a)}{x - a}$ The most common notation to denote derivative is $f'(x)$. Some other notations are $f'(x) = y'$ = $\frac{dy}{dx}$ = $\frac{d}{dx}$ $f(x)$

    The derivative of a function f(x) with respect to x is defined as  
    
$f'(x)$ = $\lim_{h \rightarrow 0}$$\frac{f(x + h) - f(x)}{h}$
There are rules for finding derivative of the functions

1. Derivative of a constant = 0
if  ƒ (x)  = c then  ƒ’(x)  = 0

2. Power rule
$ƒ(x)$ = $x^n$
then $ƒ'(x)$ = n $x^{n-1}$

3. Scalar Multiple $\frac{d}{dx}$ c $ƒ(x)$ = c $ƒ'(x)$ where c is a constant

4. Sum $\frac{d}{dx}$ (u + v) = u' + v'

5. Difference $\frac{d}{dx}$ (u - v) = $u' - v'$

6. Product rule: $\frac{d}{dx}$ ($u \times v$) = $u \times v' + v \times u'$

7. Quotient rule: $\frac{d}{dx} \frac{u}{v}$ = $\frac{v \times u' - u \times v'}{v^2}$
 
8. Reciprocal rule $\frac{d}{dx}$ ($\frac{1}{v}$) = $\frac{v'}{v^2}$  

9. chain Rule $\frac{d}{dx}$ ƒ(g(x)) = ƒ'(g(x)) $\times$ g'(x)
u and v are two different differentiable functions. → Read More Power of x

$\frac{d}{dx}$c = 0

$\frac{d}{dx}$x = 1

$\frac{d}{dx}$$x^n = n x^{n-1}$

Exponential or logarithmic functions

$\frac{d}{dx}$$e^x$ = $e^x$

$\frac{d}{dx}$$b^x$ = $b^x In(b)$

$\frac{d}{dx}$$In(x)$ = $\frac{1}{x}$

Trigonometric functions

$\frac{d}{dx}$$sin x$ = $cos x$	$\frac{d}{dx}$$csc x$ = $- csc x cot x$
$\frac{d}{dx}$$cos x$ = $-sin x$	$\frac{d}{dx}$$sec x$ = $sec x tan x$
$\frac{d}{dx}$$tan x$ = $sec^2 x$	$\frac{d}{dx}$$cot x$ = $-csc^2 x$

Inverse Trigonometric functions

$\frac{d}{dx}$$arc sin x$ = $\frac{1}{\sqrt{1 - x^2}}$	$\frac{d}{dx}$$arccsc x$ = $\frac{-1}{|x| \sqrt{x^2 - 1}}$
$\frac{d}{dx}$$arccos x$ = $\frac{-1}{\sqrt{1 - x^2}}$	$\frac{d}{dx}$$arcsec x$ = $\frac{1}{|x| \sqrt{x^2 - 1}}$
$\frac{d}{dx}$$tan x$ = $\frac{1}{1 + x^2}$	$\frac{d}{dx}$$arccot x$ = -$\frac{1}{1 + x^2}$

Derivative of fraction can be computed by applying quotient rule.
Suppose the function is $\frac{3x^2 + 1}{x^2 - 1}$ then we can apply quotient rule since it is in the form u/v where both u and v are functions of x

u = 3$x^2$ + 1

v = $x^2$ - 1

$\frac{d}{dx}$ ($\frac{u}{v}$) = $\frac{v \times u' - u \times v'}{v^2}$
 
u' = $\frac{d}{dx}$(3$x^2$ + 1) = $\frac{d}{dx}$ $3x^2$ + $\frac{d}{dx}$ 1

 = 3 $\times$ 2x + 0

u' = 6x

v = $x^2 - 1$

v' = $\frac{d}{dx}$ ($x^2 - 1$)

= $\frac{d}{dx}$ $x^2$ -$\frac{d}{dx}$ 1
 
= 2x - 0

v' = 2x

$\frac{d}{dx}$$\frac{(3x^2 + 1)}{(x^2 - 1)}$ = $\frac{(x^2 - 1)(6x) - (3x^2 + 1)(2x)}{(x^2 - 1)^2}$

$\frac{d}{dx}$ $\frac{3x^2 + 1}{x^2 - 1}$ = $\frac{6x^3 - 6x - 6x^3 - 2x}{(x^2 - 1)^2}$

$\frac{d}{dx}$ $\frac{3x^2 + 1}{x^2 - 1}$ = $\frac{-8x}{(x^2 - 1)^2}$

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