DIFFERENTIATION BY TRIGONOMETRIC SUBSTITUTION

Example 1 :

Differentiate the following

cos^-1(1 -x²)/(1+x²)

Solution :

Let y  =  cos^-1(1 -x²)/(1+x²)

x = tan θ

y  =  cos^-1(1- tan² θ)/(1+tan² θ)

y  =  cos^-1(cos 2θ)

y = 2θ

Differentiate with respect to x, we get

dy/dx  =  2 (dθ/dx)    ------(1)

x = tan θ

1  =  sec²θ (dθ/dx)

dθ/dx  =  1/sec²θ

By applying the value of dθ/dx in (1), we get

dy/dx  =  2 (1/sec²θ)  

  =  2/(1 + tan²θ)  

  =  2/(1 + x²)

Example 2 :

Differentiate the following

sin^-1(3x - 4x³)

Solution :

Let y  =  sin^-1(3x - 4x³)

x = sin θ

y  =  sin^-1(3 sin θ - 4(sin θ)³)

y  =  sin^-1(3 sin θ - 4 sin³ θ)

y  =  sin^-1(sin 3θ)

y = 3θ

Differentiate with respect to x, we get

dy/dx  =  3 (dθ/dx)    ------(1)

x = sin θ

1  =  cos θ (dθ/dx)

dθ/dx  =  1/cosθ

By applying the value of dθ/dx in (1), we get

dy/dx  =  3 (1/cos sec²θ)  

  =  2/(1 + tan²θ)  

  =  2/(1 + x²)

Example 3 :

Differentiate the following

tan^-1[(cos x + sin x) / (cos x - sin x)]

Solution :

Let y  =  tan^-1[(cos x + sin x) / (cos x - sin x)]

Divide every terms inside the parentheses by cos x, we get

y  =  tan^-1[(1 + tan x) / (1 - tan x)]

y  =  tan^-1[(tan π/4) + tan x) / (1 - (tan π/4) tan x)]

y  =  tan^-1[tan ((π/4) + x)]

y  =  (π/4) + x

Differentiate with respect to "x", we get

dy/dx  =  1

Example 4 :

Find the derivative of sin x² with respect to x²

Solution :

Let y = sin x²

here we have to differentiate with respect to "x^2" not "x"

dy/dx²  =  cosx²

Incase we differentiate with respect to "x", we get

dy/dx  =  cosx² (2x)

  =  2x cosx²

Example 5 :

Find the derivative of sin^-1(2x / (1 + x²)) with respect to tan^-1 x

Solution :

Let y  =  sin^-1(2x / (1 + x²))

 x = tan θ

y  =  sin^-1(2tan θ / (1 + tan² θ))

y  =  sin^-1(2tan θ / (1 + tan² θ))

y  =  sin^-1(sin 2θ)

y = 2θ

y = 2 tan^-1x

dy/dx  =  2 [1/(1 + x²)]

dy/dx  =  2/(1 + x²)

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