# Introduction

Let ‘f’ be a given function, then the derivative of ‘f’ is denoted by f’(x) and is defined as,

The process of finding derivative is called differentiation.

The derivative of a function can be denoted is different ways, they are
y’, y1, dy / dx etc.
The derivative of a function at ‘c’ is denoted as f’(c) and is defined as

The process of finding the derivative using definition is called the first principle of differentiation

Example: Using 1st principle of differentiation, find the derivative of (x+2)2
Let f(x) = (x + 2)2, f(x + h) = (x + h + 2)2

f’(x)

= 0 + 2x + 4
= 2(x + 2)

## Product Rule of Differentiation

If ‘u’ and ‘v’ are functions of ‘x’ then

Derivative of product of two functions is “(first function) x (derivative of second) + (second function) x (derivative of first)”
If u, v and w are functions of ‘x’ then

## Quotient Rule of Differentiation

If ‘u’ and ‘v’ are functions of ‘x’ and v≠0, then quotient rule is

Important Notes:
(i) (u ± v)’ = u’ ± v’
(ii) If a function ‘f’ is differentiable at a point ‘c’ then it is continuous                at that point.
(iii) Every differentiable function is continuous.

## Chain Rule of Differentiation

Chain Rule is applicable only for the composition of functions.  Let ‘y’ be a composition of two functions ‘f’ and ‘g’.
y = f o g = f [g(x)]
Take y= f(u) where u = g(x) so that we can find
dy/du and du/dx [Since ‘y’ is a function of ‘u’ we get dy/du and ‘u’ is a function of ‘x’ we get du/dx]
Hence

If ‘y’ is the composition of three functions ‘f’, ‘g’ and ‘u’ then
y= f o g o u
= f {g[ u(x) ]}
Take v=u(x), t=g(v) and y=f(t)
Find dv/dx, dt/dv and dy/dt

Example: Find the derivative of Cos (3x + 5)
Solution: Let y = Cos (3x + 5)
Take y = Cos (u) where u = 3x + 5

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