# DIFFERENTIATION USING CHAIN RULE EXAMPLES

## About "Differentiation Using Chain Rule Examples"

Differentiation Using Chain Rule Examples :

Here we are going to see how to find the derivatives using chain rule.

## Derivatives using chain rule - Examples

Question 1 :

Differentiate y = (x2 + 4x + 6)5

Solution :

y = (x2 + 4x + 6)5

Let u = x2 + 4x + 6

Differentiate the function "u" with respect to x, we get

du/dx  =  2x + 4(1) + 0

=  2x + 4

y = u5

Differentiate the function "y" with respect to x, we get

dy/dx  =  5u4 (du/dx)

=  5(x2 + 4x + 6)4 (2x + 4)

Question 2 :

Differentiate y = tan 3x

Solution :

y = tan 3x

Let u = 3x

Differentiate the function "u" with respect to x, we get

du/dx  =  3 (1)

=  3

y = tan u

Differentiate the function "y" with respect to x, we get

dy/dx  =  sec2u (du/dx)

=  sec23x (3)

=  3 sec23x

Question 3 :

Differentiate y = cos (tan x)

Solution :

y = cos (tan x)

Let u = tan x

Differentiate the function "u" with respect to x, we get

du/dx  =  sec2 x

y = cos u

Differentiate the function "y" with respect to x, we get

dy/dx  =  -sin u (du/dx)

=  -sin (tan x) sec2

Question 4 :

Differentiate y = ∛(1 +x3)

Solution :

y = ∛(1 +x3)

Let u = 1 +x3

Differentiate the function "u" with respect to x, we get

du/dx  =  0 + 3x2

y = u1/3

Differentiate the function "y" with respect to x, we get

dy/dx  =  (1/3) u-2/3 (du/dx)

=  (1/3) (1 + x3)-2/3 (3x2)

=  x2(1 + x3)-2/3

Question 5 :

Differentiate y = e√x

Solution :

y  e√x

Let u = √x

Differentiate the function "u" with respect to x, we get

du/dx  =  1/2√x

y = eu

Differentiate the function "y" with respect to x, we get

dy/dx  =  eu (du/dx)

=  e√x (1/2√x)

=  e√x/2√x

Question 6 :

Differentiate y = sin (ex)

Solution :

y  =  sin (ex)

Let u = ex

Differentiate the function "u" with respect to x, we get

du/dx  =  ex

y = sin u

Differentiate the function "y" with respect to x, we get

dy/dx  =  cos u (du/dx)

=  cos (ex) (ex)

=  ex cos (ex)

After having gone through the stuff given above, we hope that the students would have understood, "Differentiation Using Chain Rule Examples"

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