**Differentiation by Trigonometric Substitution :**

In this section, you will learn how to apply trigonometric identities to find derivative of a function.

**Question 1 :**

Differentiate the following

cos^{-1}(1 -x^{2})/(1+x^{2})

**Solution :**

Let y = cos^{-1}(1 -x^{2})/(1+x^{2})

x = tan θ

y = cos^{-1}(1- tan^{2} θ)/(1+tan^{2} 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 = sec^{2}θ (dθ/dx)

dθ/dx = 1/sec^{2}θ

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

dy/dx = 2 (1/sec^{2}θ)

= 2/(1 + tan^{2}θ)

= 2/(1 + x^{2})

**Question 2 :**

Differentiate the following

sin^{-1}(3x - 4x^{3})

**Solution :**

Let y = sin^{-1}(3x - 4x^{3})

x = sin θ

y = sin^{-1}(3 sin θ - 4(sin θ)^{3})

y = sin^{-1}(3 sin θ - 4 sin^{3} 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 sec^{2}θ)

= 2/(1 + tan^{2}θ)

= 2/(1 + x^{2})

**Question 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

**Question 4 :**

Find the derivative of sin x^{2} with respect to x^{2}

**Solution :**

Let y = sin x^{2}

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

dy/dx^{2 }= cosx^{2}

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

dy/dx = cosx^{2 }(2x)

= 2x cosx^{2}

**Question 5 :**

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

**Solution :**

Let y = sin^{-1}(2x / (1 + x^{2}))

x = tan θ

y = sin^{-1}(2tan θ / (1 + tan^{2} θ))

y = sin^{-1}(2tan θ / (1 + tan^{2} θ))

y = sin^{-1}(sin 2θ)

y = 2θ

y = 2 tan^{-1}x

dy/dx = 2 [1/(1 + x^{2})]

dy/dx = 2/(1 + x^{2})

After having gone through the stuff given above, we hope that the students would have understood how find derivative of a dfunction using trigonometric substitution.

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