What is Derivatives of Inverse Functions?
The derivative of an inverse function provides a way to find the rate of change of an inverse function at a point. If you have a function y = f(x) and its inverse x=f−1(y), the derivatives of these functions are related in a specific way. The details of derivative of inverse functions has been tabulated below:
Category | Subtopics | Key Points |
|---|---|---|
Derivative of an Inverse Function | Inverse Function Theorem | This theorem is fundamental for deriving the derivatives of inverse functions, showing that the derivative of an inverse function is the reciprocal of the derivative of the original function at corresponding points. |
Applying the Inverse Function Theorem | Practical application of the theorem to compute derivatives of inverse functions, illustrating step-by-step methodology. | |
Extending the Power Rule to Rational Exponents | Expands the power rule in calculus to handle cases with rational exponents, enhancing the toolkit for dealing with complex derivative problems. | |
Derivatives of Inverse Trigonometric Functions | Derivative of the Inverse Sine Function | Focuses on finding the derivative of arcsin(x), demonstrating its relationship with the derivative of the sine function. |
Applying the Chain Rule to the Inverse Sine Function | Illustrates how to use the chain rule when differentiating composite functions that include the arcsin function, crucial for more complex calculations. | |
Applying Differentiation Formulas to an Inverse Tangent Function | Details methods to differentiate the arctan function using standard differentiation formulas, useful in trigonometric calculus. | |
Applying Differentiation Formulas to an Inverse Sine Function | Similar to the inverse tangent, this points out techniques for differentiating the arcsin function using established formulas. | |
Applying the Inverse Tangent Function | Explains how to apply the arctan function in calculus problems, particularly in solving integrals and derivatives involving arctan. |
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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