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# Chain Rule

The Chain Rule, also called the composite function, is applicable to a function that is made up of another function. For example –

y = (2x + 3)2 can be treated as a composite of two functions.

Let f(x) = 2x + 2  = u

Then y = u2

Therefore  =

This way of differentiation is called the Chain Rule.

Now differentiating our function using this rule, we get

u = 2x + 3                                 y = u2

= 2                               = 2u

=

= 2u x 2  = 4u

= 4(2x + 3)

= 8x + 12

In general,   = g’f(x)  x  f'(x),  and

=  nf'(x)

Now let us look at a few examples. Differentiate the following functions:

1. y = 3 (5x + 4)4

=  nf'(x)

y’  = 3×4 (5x + 4)3 x 5  =  60(5x + 4)3

2.  y =  (x4 – 2x3 + 5x + 2)2

y’ = 2.(x4 – 2x3 + 5x + 2)1.(4x3 – 6x2 + 5)

= (2x4 – 4x3 + 10x + 4).(4x3 – 6x2 + 5)

3. y =

y’ =

=

=

4. Find the equation of the tangent to y = (2x + 3)4 at the point x = -1

When x = -1, y = (2x-1 + 3)4 = (-2 + 3)4  =  1   (-1, 1)

y = (2x + 3)4

= 4.(2x + 3)3.2    =  8(2x + 3)3

at x = -1  =  m = 8(2 x -1 + 3)3  =  8

Equation of the tangent at (-1, 1) is

(y – 1)  =  8(x + 1)

8x – y + 8 + 1 = 0

8x – y + 9  = 0