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 =
x
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
=
x
= 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:
-
y = 3 (5x + 4)4
= nf'(x)
y’ = 3×4 (5x + 4)3 x 5 = 60(5x + 4)3
-
y = (x4 – 2x3 + 5x + 2)2
y’ = 2.(x4 – 2x3 + 5x + 2)1.(4x3 – 6x2 + 5)
= (2x4 – 4x3 + 10x + 4).(4x3 – 6x2 + 5)
-
y =
y’ =
=
=
-
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