DIFFERENTIATION 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) sec²x 

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 = u^1/3

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 = e^u

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

dy/dx  =  e^u (du/dx)

=  e^√x (1/2√x)

=  e^√x/2√x

Question 6 :

Differentiate y = sin (e^x)

Solution :

y  =  sin (e^x)

Let u = e^x

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

du/dx  =  e^x  

y = sin u

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

dy/dx  =  cos u (du/dx)

=  cos (e^x) (e^x)

=  e^x cos (e^x)

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