Solved Example on Derivative of Inverse Function
Example 1. Find the derivative of y = tan -1 (x2).
Solution:
Differentiating both side, we get
[Tex]\frac{dy}{dx} = \frac{d(tan^{-1} (x^2))}{dx} [/Tex]
Using the inverse derivative of tan-1 θ
= [Tex]\frac{1}{1+(x^2)^2} \frac{d (x^2)}{dx} [/Tex]
= [Tex]\frac{2x}{1+x^4}[/Tex]
Example 2. Find the derivative of y = sin -1 (3x-2).
Solution:
Differentiating both side, we get
[Tex]\frac{dy}{dx} = \frac{d (sin^{ -1} (3x-2))}{dx} [/Tex]
Using the inverse derivative of sin-1 θ
= [Tex]\frac{1}{\sqrt{1-(3x-2)^2}} \frac{d(3x-2)}{dx} [/Tex]
= [Tex]\frac{1}{\sqrt{1-(9x^2-18x+4)}} [/Tex](3)
= [Tex]\frac{3}{\sqrt{(18x-9x^2-4)}}[/Tex]
Example 3. Find the derivative of y = cos -1 (1 – x2).
Solution:
Differentiating both side, we get
[Tex]\frac{dy}{dx} = \frac{d(cos^{ -1} (1 – x^2))}{dx} [/Tex]
Using the inverse derivative of cos-1 θ
= [Tex]\frac{-1}{\sqrt{1-(1-x^2)^2}} \frac{d(1-x^2)}{dx} [/Tex]
= [Tex]\frac{-1}{\sqrt{1-(1-2x^2+x^4)}} [/Tex](-2x)
= [Tex]\frac{2x}{\sqrt{(2x^2-x^4)}}[/Tex]
= [Tex]\frac{2}{\sqrt{(2-x^2)}}[/Tex]
Derivatives of Inverse Functions
In mathematics, a function(e.g. f), is said to be an inverse of another(e.g. g), if given the output of g returns the input value given to f. Additionally, this must hold true for every element in the domain co-domain(range) of g. E.g. assuming x and y are constants if g(x) = y and f(y) = x then the function f is said to be an inverse of the function g. Or in other words, if a function f : A ⇢ B is one – one and onto function or bijective function, then a function defined by g : B ⇢ A is known as inverse of function f. The inverse function is also known as the anti function. The inverse of function is denoted by f-1.
f(g(x)) = g(f(x)) = x
Here, f and g are inverse functions.
Table of Content
- Overview of Derivatives of Inverse Functions
- Procedure of finding inverse of f
- Derivatives of Inverse Functions
- How to find derivatives of inverse functions from the table?
- Derivatives of Inverse Trigonometric Functions
- How to find the derivatives of inverse trigonometric functions?
- 1. Derivative of f given by f(x) = sin–1 x.
- 2. Derivative of f given by f(x) = cos–1 x.
- 3. Derivative of f given by f(x) = tan–1 x.
- 4. Derivative of f given by f(x) = cot–1 x.
- 5. Derivative of f given by f(x) = sec–1 x.
- 6. Derivative of f given by f(x) = cosec–1 x.
Contact Us