Derivatives of Inverse Functions
What is the derivative of an inverse function?
The derivative of an inverse function, f−1(y), provides the rate of change of the inverse function at a particular point. It is given by the reciprocal of the derivative of the original function f(x) evaluated at the corresponding point.
What is the formula for the derivative of an inverse function?
The formula of derivative of inverse function is (f−1)′(y) = [Tex] \frac{1}{f'(x)}[/Tex], where x=f−1(y)
What conditions must a function meet for its inverse to be differentiable?
For a function f to have a differentiable inverse f−1:
- f must be continuous.
- f must be one-to-one (bijective), ensuring it has an inverse.
- f must be differentiable with a non-zero derivative f′(x)≠0 for all x in its domain
What happens if the derivative of the original function is zero at a point?
If f′(x)=0 at a point x, then the inverse function is not differentiable at the corresponding point y=f(x), because you cannot divide by zero
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.
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