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

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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)

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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

• First Principles 
• Implicit Function
  
• Parametric Function
• Substitution Method
  
• logarithmic function
  
• Product Rule
  
• Quotient Rule
• Rate of Change
• Rolle's theorem
• Lagrange's theorem
• Finding increasing or decreasing interval
  
• Increasing function
  
• Decreasing function
• Monotonic function
• Maximum and minimum
• Examples of maximum and minimum

Chain Rule Of Differentiation to First Principles