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 = u^{2}
Therefore = x
This way of differentiation is called the Chain Rule.
Now differentiating our function using this rule, we get
u = 2x + 3 y = u^{2}
= 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 = (x^{4} – 2x^{3} + 5x + 2)^{2}
y’ = 2.(x^{4} – 2x^{3} + 5x + 2)^{1}.(4x^{3} – 6x^{2} + 5)
= (2x^{4} – 4x^{3} + 10x + 4).(4x^{3} – 6x^{2} + 5)

y =
y’ =
=
=

Find the equation of the tangent to y = (2x + 3)^{4} at the point x = 1
When x = 1, y = (2x1 + 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