The derivative is a measure of how a function changes as its input changes. Derivative is also the rate of change of function(f) with respect to variable(x), as the variable changes from x to x + h (where h approaches to zero). The process of finding a derivative is called differentiation. We shall study in detail the meaning and the process of differentiation. Its found that the concept of differentiation has been applied to many other fields as well. This section will help us to understand the concept of differentiation.

Theorem : Derivative of a Constant is zero.

$\frac{d}{dx}$$(c)$ = $0$, for all x $\in$ R

Proof
: Let $f(x)$ = $c$, for all x $\in$ R where c is a fixed number.

Then $f(x+\delta x)$ = $c$

By differentiating $f'(x)$ = $\lim_{\delta x \to 0}$ $\frac{f(x+\delta x)-f(x)}{\delta x}$

= $\lim_{\delta x \to 0}$ $\frac{c-c}{\delta x}$

= $\frac{0}{\delta x}$

= $\lim_{\delta x \to 0}0$

= $0$

Example: Derivative of $1$

$\frac{d}{dx}$$(1)$ = $0$
$\frac{x^{x}}{dx}$ = $(\log x +1)x^{x}$

Proof: Let $y$ be the given function

i.e. $y$ = $x^{x}$

Take logarithm on both sides
$\log y$ = $\log x^{x}$

= $x\log{x}$

Differentiating the above equation with respect $x$.

$\frac{1}{y} \frac{dy}{dx}$ = $\log x+x$$\frac{1}{x}$

= $\log x +1$

$\therefore$$\frac{dy}{dx}$ = $(\log x +1)y$

$\frac{dy}{dx}$ = $(\log x +1)x^{x}$

Derivative of a^x = $\frac{d}{dx}a^{x}$ = $\log{a}.a^{x}$

Example:

Derivative of $5^{x}$ = $\frac{d}{dx}5^{x}$ = $\log{5}.5^{x}$

Derivative of $3^{x}$ = $\frac{d}{dx}3^{x}$ = $\log{3}.a^{3}$

Derivative of $2^{x}$ = $\frac{d}{dx}3^{x}$ = $\log{3}.a^{3}$

1. Derivative of $x^{n}$, $n$ is rational number


let $f(x)$ = $x^{n}$ then $f(x+\delta x)$ = $(x+\delta x)^{n}$

$\therefore$ by differentiation $f '(x)$ = $\lim_{\delta x \to 0}$ $\frac{f(x+\delta x)-f(x)}{\delta x}$

= $\lim_{\delta x \to 0}$ $\frac{(x+\delta x)^{n} - (x)^{n}}{\delta x}$

= $\lim_{x+\delta x \to x}$ $\frac{(x+\delta x)^{n} - (x)^{n}}{(x+\delta x)-x}$ $(\because \delta x \to
0 \Rightarrow x+\delta x \to x)$

= $n.x^{n-1}$ (using $\lim_{x \to a}$ $\frac{x^{n}-a^{n}}{x-a})$ = $na^{n-1}$

i.e. $\frac{d}{dx}(x^{n})$ = $n.x^{n-1}$ , This is known as Power Formula.


2. Derivative of $(ax+b)^{n}$, $n$ is rational number


$\frac{d}{dx}$$((ax+b)^{n})$ = $n(ax+b)^{n-1}. a$

Below is some solved examples:

Solved Examples

Question 1: Derivative of $x$
Solution:
 
$\frac{d}{dx}(x)$ = $\frac{d}{dx}$ $x^{1}$
= $1. x^{1-1}$
= $1x^{0}$
= $1.1$
= $1$
 

Question 2: Derivative of $x^{2}$
Solution:
Derivative of $x^{2}$= $\frac{d}{dx}$ $x^{2}$
= $2. x^{2-1}$
= $2x^{1}$
= $2.x$
= $2x$

Question 3: Derivative of $2x$
Solution:
 
Derivative of $2x$ = $\frac{d}{dx}$ $2x$

= $2$$\frac{d}{dx}x$

= $2(1.x^{1-1})$
= $2x^{0}$
= $2.1$
= $2$
 

Question 4: Derivative of $\frac{1}{x}$
Solution:
 
Derivative of $\frac{1}{x}$

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

= $-1. x^{-1-1}$ = $2x^{-2}$

= $2.$ $\frac{1}{x^{2}}$

= $\frac{2}{x^{2}}$
 

Question 5: Find the derivative of the function $\sqrt{3x-5}$
Solution:
Let $y$ = $\sqrt{3x-5}$ =$(3x-5)^{\frac{1}{2}}$

Diffr. Wit respect to $x$ we will get

$\frac{dy}{dx}$ = $\frac{1}{2}$$(3x-5)^{\frac{1}{2}-1}$.$3$

= $\frac{3}{2} \frac{1}{\sqrt{3x-5}}$

Question 6: If $f(x)$ =$x^{n}$ and $f'(1)$ = $10$, find the value of n?
Solution:
 
Given $f(x)$ =$x^{n}$

$\therefore$ $\frac{d}{dx}(x^{n})$ = $f'(x)$ = $n.x^{n-1}$

$f'(1)$ = $n.1^{n-1}$ = $n$

but $f'(1)$ = $10$

$n$ = $10$