chain rule of differentiation

Chain Rule Of Differentiation

In this page chain rule of differentiation we are going to see the one of the method using in differentiation.We have to use this method when two functions are interrelated.Now let us see the example problems with detailed solution to understand this topic much better.

Example 1:

Differentiate log√x with respect to x

Solution:

we have formula only for differentiating log x we don't have formula for differentiating log √x. To differentiate this problem we have to use chain rule.

Let y = log √x

we are going to take u = √x.Now the given function becomes y = log u

let us write the formula to solve this problem.Here the function "y" is defined by he variable "u" and the variable "u" is defined by "x"

dy/dx = (dy/du) x (du/dx)

dy/du = 1/u

du/dx = 1/(2√x)

dy/dx = (1/u) x 1/(2√x)

= 1/[(2√x) u]

= 1/[(2√x)√x ]

dy/dx = 1/2x

Example 2:

Differentiate sin (ax+b) with respect to x

Solution:

we have formula only for differentiating sin x we don't have formula for differentiating sin (ax+b). To differentiate this problem we have to use chain rule of differentiation.

let y = sin (ax + b) and we are going to take u = ax + b

Now the function becomes y = sin u

let us write the formula to solve this problem.Here the function "y" is defined by he variable "u" and the variable "u" is defined by "x"

dy/dx = (dy/du) x (du/dx)

dy/du = cos u

du/dx = a(1) + 0

du/dx = a

dy/dx = (cos u) x a

Now we need to replace u by its value ax + b

dy/dx = cos (ax + b) a

= a cos (ax + b)

Example 3:

Differentiate log (sin x) with respect to x

Solution:

we have formula only for differentiating log x we don't have formula log (sin X) for differentiating.To differentiate this problem we have to use chain rule.

let y = log (sin X) and we are going to take u = sin X

Now the function becomes y = log u

let us write the formula to solve this problem.Here the function "y" is defined by he variable "u" and the variable "u" is defined by "X"

dy/dx = (dy/du) x (du/dx)

dy/du = 1/u

du/dx = cos X

dy/dx = (1/u) x cos X

= cos X/u

= cos X/sin X

= cot X