Inverse Derivative Calculator

Introduction to the Inverse Derivative Calculator:

Inverse derivative calculator is an online tool that helps you to evaluate the differentiation of an inverse function in a few seconds. It is used to find the derivative of a function and the inverse of that derivative function in a few seconds.

It is a beneficial tool for students, teachers, and researchers as it gives you solutions of complex inverse derivative functions because it is used in various fields like engineering, physics, financial analysis to estimate the risk assessment in scientific research where you can analyze experimental data or theoretical models.

What is Inverse Derivative?

The inverse derivative of a function refers to the derivative of the inverse of a given function. It is denoted as f⁻¹(x). If you have a function f(x) then its inverse function f⁻¹(x), is the function that is the function of f(x). This means that f(f⁻¹1(x))=x and f⁻¹(f(x))=x for all x in the domain of f and f⁻¹.

There's a specific relationship between the derivatives of a function f(x) and its inverse f⁻¹(x). If y=f(x), then x=f⁻¹(y). The derivatives of this types of functions are given as:

$$ \frac{dy}{dx} \;=\; \frac{d}{dx}(f^{-1}(x)) \;=\; (f^{-1})’ (x) \;=\; \frac{1}{f’ (f^{-1}(x))} $$

How to Calculate the Derivative of an Inverse Function?

For the calculation of the derivative of an inverse function you need to find the relationship between a function f(x) and its inverse f⁻¹(x). Here is a calculation step to find (f⁻¹)′(x) which is given as:

Step 1:

Identify the original function y=f(x) to determine its inverse x=f⁻¹(y)

Step 2:

Differentiate the function on both sides of the equation with respect to y.

$$ \frac{d}{dy}(x) \;=\; \frac{d}{dy}(f^{-1} (y)) $$

Since x=f^−1(y), then when you differentiate x with respect to y it gives the solution in the form of,

$$ \frac{dx}{dy} \;=\; \frac{d}{dy}(f^{-1}(y)) $$

Step 3:

The formula for the derivative of an inverse function f⁻¹(x),

$$ (f^{-1})’ (y) \;=\; \frac{1}{f’(f^{-1}(y))} $$

Step 4:

Put f⁻¹(y) back into the above formula as,

$$ \frac{dx}{dy} \;=\; \frac{1}{f’ (f^{-1} (y))} $$

Step 5:

Lastly, shows the derivative in terms of x such that x = f⁻¹(y) as,

$$ \frac{dx}{dy} \;=\; \frac{1}{f’ (x)} $$

Solved Example of Inverse Derivative:

The derivative of inverse function calculator helps you to solve the inverse derivative problem. But it is important to understand the manual calculations. So an example is given below,

Example: Use the inverse function theorem to find the derivative of,

$$ g(x) \;=\; \frac{x + 2}{x} $$

Solution:

The inverse of:

$$ g(x) \;=\; \frac{x + 2}{x}\; is\; f(x) \;=\; \frac{2}{x - 1} $$

Take the derivative of the given function f(x) with respect to x,

$$ f’(x) \;=\; \frac{-2}{(x - 1)^2} $$

Find the inverse derivation of a function which is,

$$ f’ (g(x)) \;=\; \frac{-2}{(g(x))^2} \;=\; \frac{-2}{\left( \frac{x + 2}{x} - 1 \right)^2} \;=\; -\frac{x^2}{2} $$

The inverse of a given function is equal to,

$$ g’(x) \;=\; \frac{1}{f’ (g(x))} \;=\; -\frac{2}{x^2} $$

Hence the solution of a given inverse function is,

$$ g’(x) \;=\; -\frac{2}{x^2} $$

How to Use the Derivative of Inverse Function Calculator?

The derivative of the inverse calculator has an easy-to-use interface, so you can easily use it to evaluate the given inverse differential function solution. Before adding the input for the solutions of given function problems, you must follow some simple steps. These steps are:

1. Enter the inverse derivation function in the input field that you want to evaluate using the differential rule.
2. Add the derivative variable on which your inverse function is differentiated in the input field.
3. Recheck your input value of the inverse function problem before hitting the calculate button to start the calculation process in the derivative inverse calculator.
4. Click on the “Calculate” button to get the desired result of your given derivation inverse function problem.
5. If you want to check the accuracy of our inverse derivative calculator in the solution, then use the load example.
6. Click on the “Recalculate” button to get a new page for solving more inverse derivative questions.

Final Result of Derivative of the Inverse Calculator:

The derivative of inverse function calculator gives you the solution when you add the input value to it. It provides you with solutions that may contain as:

• Result Option:

The result option provides you with a solution to inverse derivative questions.

• Possible Step:

When you click on the possible steps option it provides you with the solution of the inverse differential problem with steps.

• Plot Option:

Plot option provides a solution of inverse function graphs for visual understanding.

Advantages of Derivative of Inverse Functions Calculator:

Derivative inverse calculator gives you multiple benefits whenever you use it to calculate inverse function problems to get its solution. These benefits are:

• Our tool saves the time and effort that you consume in solving inverse differential questions to get solutions in a few seconds.
• It is a free-of-cost tool that provides you with a solution for a given inverse derivation problem to find the inverse function f(x) using the differential rules of calculus without spending.
• Inverse derivative Calculator will give you accurate results when you are computing the inverse function f⁻¹(x) but in the manual calculation, you may commit mistakes.
• It is an adaptive tool that allows you to find the different types of inverse derivation like whether it is the logarithmic, exponential, or trigonometric function in this calculator.
• You can use this derivative of the inverse calculator for practice to get familiar with this concept of the inverse derivative function.
• The derivative of inverse function calculator is a trustworthy tool that provides you with correct solutions as per your input to calculate the inverse derivative problem.