Derivative of Inverse Trignometric Functions

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Before finding the differentiation of inverse trigonometric functions, recall how the inverse trigonometric functions are defined and what the domain and range of each inverse trigonometric function. For ready reference, the domain and range of these functions are tabulated below.Before finding the differentiation of inverse trigonometric functions, recall how the inverse trigonometric functions are defined and what the domain and range of each inverse trigonometric function. For ready reference, the domain and range of these functions are tabulated below.

Whenever we say differentiability of these functions we consider them in their respective domains.

Derivative of sin^-1x

Let y = sin^-1 x, then sin y = x.

Derivative of cos^-1x

Differentiating both sides with respect to x, we get

Hence, we have

Derivative of tan^-1x

Let y = tan^-1x. Then tan y = x.

Differentiating both sides w.r.t. x, we get

Derivative of cot^-1x

Differentiating both sides with respect to x, we get

Hence, we have

Derivative of sec^-1x

Let y = sec^-1x. Then sec y = x.

Differentiating w.r.t. x, we have

Similarly,

Example 1:

Suggested answer:

Let

Differentiating with respect to x, we get

Example 2:

Differentiate cos^-1(2x+3) from first principles.

Suggested answer:

Let y = cos^-1(2x+3)

2x + 3 = cos y ……….(1)

Let Dx be an increment of x and Dy be the corresponding increment of y.

Therefore 2(x + Dx) + 3 = cos (y + Dy)

2x + 2Dx + 3 = cos (y + Dy) ……….(2)

Subtracting (1) from (2), we have

2x + 2Dx + 3 - 2x - 3 = cos (y + Dy) - cosy

Using the formula,