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DIFFERENTIATION

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Introduction

differentiation8




Let ‘f’ be a given function, then the derivative of ‘f’ is denoted by f’(x) and is defined as, differentiation9




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
differentiation10

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)differentiation11

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

List of derivatives of certain standard functions

differentiation12

Product Rule of Differentiation

If ‘u’ and ‘v’ are functions of ‘x’ then
differentiation13

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
differentiation14

Quotient Rule of Differentiation

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

differentiation15

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 differentiation17

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

                      differentiation18

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

                differentiation19

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