Derivative of Inverse Trigonometric Functions: Overview, Questions, Preparation

Inverse trigonometric functions are often referred to as arcus functions, anti-trigonometric functions, or cyclometric functions. These functions are often used to produce an angle for a trigonometric value. Inverse trigonometric functions have diverse uses in engineering, geometry, navigation, etc.

sin–1 : [–1, 1] → [-π/2, π/2]    

cos –1 : [–1, 1] → [0, π] 

cosec–1 : R – (–1,1) → [-π/2, π/2]  – {0} 

sec –1 : R – (–1, 1) → [0, π] – { π/2 } 

tan–1 : R → [-π/2, π/2]     

cot–1 : R → (0, π)

(i) sin–1 1/x = cosec–1 x, x ≥ 1 or x ≤ – 1 

(ii) cos–1 1/x = sec –1 x, x ≥ 1 or x ≤ – 1

(iii) tan–1 1/x = cot–1 x, x > 0

(iv) sin–1 (–x) = – sin–1 x, x ∈ [– 1, 1] 

(v) tan–1 (–x) = – tan–1 x, x ∈ R 

(vi) cosec–1 (–x) = – cosec–1 x, | x | ≥ 1

(vii) cos–1 (–x) = π – cos–1 x, x ∈ [– 1, 1] 

(viii) sec–1 (–x) = π – sec–1 x, | x | ≥ 1 

(ix) cot–1 (–x) = π – cot–1 x, x ∈ R

(x) sin–1 x + cos–1 x = π/2 , x ∈ [– 1, 1] 

(xi) tan–1 x + cot–1 x = π/2 , x ∈ R 

(xii) cosec–1 x + sec–1 x = π/2 , | x | ≥ 1

(xiii) tan–1 x + tan–1 y = tan–1 x + y / 1– xy , xy

(xiv) tan–1 x – tan–1 y = tan–1 x – y/1 + xy , xy > – 1

(xv) 2tan–1 x = sin–1 2x/1+x2, | x | ≤ 1 

(xvi) 2tan–1 x = cos–1 1-x2/1+x2, x ≥ 0 

(xvii) 2 tan–1 x = tan–1 2x/1-x2, – 1

Example:

Differentiate the function f(x) = cos-1x using the first principle.

limh->0 {f(x + h) – f(x)} / h

cos-1x + sin-1x = pi/2

cos-1x = pi/2 – sin-1x

f(x) = cos-1x

f(x + h) = cos-1(x + h)

limh->0 {cos-1(x + h ) – cos-1(x)} / h

limh->0 {pi/2 – sin-1(x + h) – (pi/2 – sin-1x) } / h

limh->0 {pi/2 – sin-1(x + h) – pi/2 + sin-1x } / h

– limh->0 {sin-1(x + h) – sin-1x} / h

limh->0 { sin-1(x + h) – sin-1x } / h = 1 / √(1 – x2)

– 1 / √(1 – x2)

This concept is taught under the chapter Derivative of Inverse Trigonometric Functions

in class 12. In this chapter, you will learn about the nature of inverse trigonometric functions and their derivatives and use this knowledge to solve questions. The weightage of this chapter is four marks. 

1. Find y’ if y=arctanx3

2. Find f′( x) if f( x) = cos −1(5 x).

f’(x) = -1/1-(5x)2 . 5

= -5/1-25x2

3. Differentiate y=5x6−sec−1(x)

dy/dx=30x5−1/x√x2−1

Q: What are the six inverse trig functions?

Q: What is the derivative of an inverse function?

Q: What is the derivative of tan inverse?

Q: Is arctan inverse tan?

Q: What is the formula for derivatives?