## 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.

**Online Tutoring for SAT Math by Expert Indian Math Tutor**

**(Based on The College Panda's SAT Math : Advanced Guide and Workbook for the New SAT)**

**Tutoring Charges $10 per hour **

**No credit card required**

**Free demo for 30 minutes**

**Mail us : v4formath@gmail.com **

**Skype id : rams_tpt**

**Mobile : +919994718681**

**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

**Related Topics **

Chain Rule Of Differentiation to First Principles