# Chain Rule Calculator

## Introduction to Chain Rule Calculator

A multivariable chain rule calculator is an online free tool that displays the derivative of a given function. It calculates the derivative of a function by using the chain rule.

The Chain rule is a rule in calculus for finding derivatives of the compositions of two or more functions.

While computing derivatives of the composition of two or more functions, the process is so complicated that you can forget to differentiate one or more terms of the function. To avoid this kind of complication, introduce an online tool that is capable of calculating derivatives of the composition of functions.

## What is a Chain Rule Derivative Calculator?

The derivative chain rule calculator is an online tool that is capable of finding derivatives for a composition of functions. It follows the differentiation rules and finds the product within a few moments. And the solution will be step by step in simplified form.

Since the chain rule derivative calculator follows the chain rule formula for differentiation, it collects the function from the user and provides a possible solution for it. You need to write the value of the function in a simple form because the chain rule differentiation calculator does not require a specific software (i.e., math type) to write the function.

## How to find derivatives using chain rule multivariable calculator?

Let’s see how this tool quickly calculates derivatives for the composition of functions than the manual solution.

There are elementary steps to perform derivation for a function. Follow these steps to find a solution using the chain rule derivative calculator:

1. In the first step you have to write the function in the “Enter Function” box.
2. In the second step, choose the variable by which you want to calculate the derivative of the given function. Which can be selected from the “With respect to” box.
3. In the third and last step, click on the “Calculate” button.

After clicking on calculate, you will find the final derivative and all the steps leading to the answer.

## Formula used by chain rule differentiation calculator

If f and g are differentiable functions of a single variable and function F is defined by for all x,

$$F'(x)\;=\;f'(g(x))g'(x)$$

Simplest form is,

$$\frac{dy}{dx}=\frac{d}{dx}{f(g(x))}\;=\;f'(g(x))g'(x)$$

The chain rule multivariable calculator uses following formula to calculate derivative of a given function:

$$\frac{dy}{dx}\;=\;\frac{dy}{dz}\times\frac{dz}{dx}$$

Where y = f(z) = f(g(x)) and z = g(x)

Let’s understand this rule with an example.

Consider y=z1/2 and z=x2+x. We have to find dy/dx.

In first step,

$$\frac{dz}{dx}\;=\;\frac{d}{dx}(x^2+x)$$

Now,

$$\frac{dz}{dx}\;=\;2x\;+\;1$$

Similarly,

$$\frac{dy}{dz}\;=\;\frac{d}{dz}(z^{\frac{1}{2}})$$

Using power rule of derivative,

$$\frac{dy}{dz}\;=\;\frac{1}{2}z^{\frac{1}{2}-1}\frac{d}{dz}(z)$$

We will get,

$$\frac{dy}{dz}\;=\;\frac{1}{2}z^{\frac{1}{2}}$$

More simplification,

$$\frac{dy}{dz}\;=\;\frac{1}{2z^{\frac{1}{2}}}$$

To find dy/dx, we have to choose chain rule. So,

$$\frac{dy}{dx}\;=\;\frac{dy}{dz}\;\times\;\frac{dz}{dx}$$

Putting the values of dy/dz and dz/dx, we get

$$\frac{dy}{dx}=\frac{1}{2(z^\frac{1}{2})}\;\times\;2x\;+\;1$$

Since z =x2+x. Substituting the value of z in the above equation. Then,

$$\frac{dy}{dx}\;=\;\frac{1}{2(x^2\;+\;x)^\frac{1}{2}}\times\;2x\;+\;1$$

Here is the derivation by using the chain rule. We have also seen that manual differentiation using the chain rule is lengthy. But when the same example is given to the chain rule differentiation calculator, it shows results very quickly. It does not show results only but gives possible derivations, roots, and domain and plots a graph as a visual representation.

## Why to use Chain Rule Calculator?

The chain rule is a vital differentiation rule in calculus and differential geometry. To learn and understand the concept of derivative, you have to practice all of its rules like the power rule, product rule, quotient rule, and chain rule.

Sometimes, it isn't easy to differentiate a function with more than two variables by hand. There you need a device or a tool that can help you do that.

When calculating derivatives by using the chain rule, it is possible that somewhere you can forget to apply the rule. This is because the derivation is so lengthy. So, this is why you must use a chain rule multivariable calculator.

## Benefits of using Chain Rule Calculator

The chain rule derivative calculator is beneficial for students to calculate derivatives of those functions in which 2 more variables are included. There are some advantages of using the chain rule differentiation calculator:

1. Instead of wasting time-solving lengthy derivations, the calculator helps you find results quickly and correctly.
2. It gives you a step-by-step solution so that the user can easily understand the method.
3. It is a chain rule multivariable calculator. You can easily calculate the derivative of a function that has a composition of two or more variables.
4. The calculator is reliable because it works quickly and efficiently.
5. You can get a visual representation function in a graph.
6. You don’t have to pay any fee for using the chain rule multivariable calculator.
7. It also calculates the root for the given function.
8. It verifies the answer by integrating it.
9. It tells whether the domain of the given function is actual or not.
10. The online tool first differentiates the function and then determines it using the chain rule.

### Alan Walker

Last Updated June 02, 2022

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.