Chain Rule Calculator

Getting bored from solving chain rule questions again and again? No worries as our chain rule calculator will help you solve the chain rule function in seconds.


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Introduction to Chain Rule Calculator with Steps

Chain Rule Derivative Calculator is an online multivariable tool for derivatives that provides the chain rule of derivative value to a given composition function. Our composite function derivative calculator makes the calculation faster, which gives the differentiation solution within a fraction of a second.

A chain rule calculator is a tool that automates the process of finding derivatives using the chain rule. Users can input a composite function, and the calculator will compute its derivative step by step, applying the chain rule where necessary.

chain rule calculator

Our calculator serves as a valuable tool for students, professionals, and anyone working with functions in calculus, helping them understand and compute derivatives efficiently and accurately.

Additionally, if you would like to find the derivative of product of two function, you can be use our derivative calculator product rule. Our calculator also help to evaluate derivatives of products of two or more than two functions using this product rule calculator with steps in a fraction of a second.

What is the Chain Rule in Derivation?

Chain rule is a method for determining the derivation of composite functions. It is applied to the number of functions that are made up of the composition. Therefore, the chain rule provides the formula to calculate the derivative of a composition of functions.

The chain rule is applied when dealing with composite functions, where one function is nested within another. Additionally, if you would like to find the derivative of a function that is present in the form of the ratio of two functions, you can utilize our derivative quotient calculator.

Our calculator is highly beneficial for students, teachers, and professionals who want to solve calculus problems related to quotient rules efficiently.

Formula Used by the Chain Rule Derivative Calculator

Chain rule is a technique to differentiate a function that has different independent variables.

$$ \frac{dy}{dx} \;=\; \frac{dy}{dt} . \frac{dt}{dx} $$

Here, dy/dx shows the y derivative with respect to x.

dy/dt shows the derivative of y with respect to t.

dt/dx shows the derivative of t with respect to x

This formula states that the derivative of the product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

Realted:For further exploration and to solve complex calculus problems, you can use our l'hopital's rule calculator. This calculator applies L'Hopital's rule to evaluate limits involving indeterminate forms, providing step-by-step solutions for enhanced understanding.

Working Method of the Chain Rule Calculator

The composite derivative calculator helps users to find the various types of the composition of differentiable functions because all differential chain rules have been installed on its server.

When you give chain rule derivative problems as input in this derivative of composite function calculator, it will first identify the nature of a given function. After analyzing the first and second functions, the differentiation process starts, where the first function is differentiated concerning a variable by keeping the second function constant. Then it takes the derivative of the second function.

At last, the chain rule solver multiplies the first and second and gives the solution to a given problem. It is one of the most accurate calculators you will find on an online platform regarding reliability. You can find the graph using this chain rule derivative calculator.

Further, If you're interested in exploring other calculus-related calculator, you can go through our euler's method calculator. Our calculator also allows you to approximate solutions to ordinary differential equations using Euler's method, providing a helpful resource for numerical analysis in calculus.

Let us see an example to understand the working process of a composite function derivative calculator.

Solved Example of the Chain Rule

An example of a chain rule derivative with the solution is given below:


$$ \frac{d}{dx} \biggr[ cos(x)^3 * sin(t) \biggr] $$


$$ \biggr[ \frac{\partial }{\partial x} (cos^3(x) \biggr] sin(t) $$

Factor out constants:

$$ \biggr[ \frac{\partial}{\partial x} (cos^3 (x) \biggr] sin(t) $$

Using the chain rule,∂/∂x (cos3 (x)) ∂/∂u ∂u/∂x, where u = cos(x) and ∂/∂u (u3) = 3u2:

$$ 3cos^2 (x) \biggr[ \frac{\partial}{\partial x} (cos(x)) \biggr] sin(t) $$

Using the chain rule, ∂/∂x (cos(x)) = ∂cos(u)/∂u ∂u/∂x, where u = x and ∂/∂u (cos(u)) = -sin(u):

$$ -\biggr[ \frac{\partial}{\partial x} \biggr] sin(x) 3cos^2 (x) sin(t) $$

Simplify the expression:

$$ -3cos^2 (x) \biggr[ \frac{\partial}{\partial x} (x) \biggr] sin(t) sin(x) $$

The derivative of x is 1:

$$ -3cos^2 (x) sin(t) sin(x) 1 $$

Simplify the expression:

$$ -3 cos^2 (x) sin(t) sin(x) $$

Thus it is the final solution of our function with specific limits. For further exploration and to solve numerical problems efficiently, you can utilize our newton's method calculator. This calculator applies Newton's method to find successively better approximations to the roots of a real-valued function.

How to Use the Composite Function Derivative Calculator?

Chain rule calculator with steps is an easy-to-use tool for finding complex chain rule differential problems in no time if you follow the given assistance as a guideline. These steps are:

  1. Enter the function f(x).g(y) format in the input box.
  2. Select the differential variable from the list of (x,y,z,t,u) etc
  3. Review your function before pressing the calculate button.
  4. Click on the "Calculate" button, the results will display on the screen.
  5. For more calculations click on the “Recalculate” button.

Reelated:If you're interested in exploring other mathematical calculators, you can use our improved euler method calculator. This calculator assists in approximating solutions to ordinary differential equations using the improved Euler method, providing another valuable tool for numerical analysis.

Composite Derivative Calculator's Results:

After giving the input, the derivative of composite function calculator will give you the solution of the given function with some additional steps. It may contain:

Derivative solution along with given derivative problem.

In the next box, possible immediate steps give a complete solution to chain rule problems.

Then comes the plot option where you can get a graphical representation of a given function.

Why Do You Need a Chain Rule Solver?

In calculus, derivation is difficult to evaluate especially when it comes to chain rules. If you are manually practicing and have good concepts, then you can deal with it easily. Otherwise, you need some external help like our chain rule calculator to solve chain rule equations.

It provides step-by-step results so that a user can understand this concept more effectively. You can use a lot of different examples to calculate and understand.

The chain rule derivative calculator will be very crucial for your overall learning of the chain rule concept of derivation.

Benefits of the Derivative of Composite Function Calculator

The chain rule calculator is a useful tool to find the composition of complicated differentiable functions by using a chain rule of differentiation, as it gives a lot of benefits which are:

  1. This composite derivative calculator saves the time that you spend on manual calculations.
  2. It has a user-friendly interface so everyone can access it.
  3. Our chain rule solver provides you with accurate results.
  4. It is free of cost to find the composition of differentiable functions using this calculator.

By utilizing the Chain Rule Solver, you can streamline the process of solving chain rule equations, gain insight into complex derivative problems, and strengthen your overall understanding of calculus concepts.

Additionally, for further exploration and to solve numerical problems efficiently, you can use our bisection method calculator. This calculator applies the bisection method to find roots of a real-valued function within a specified interval.

Take Care of these Errors While Using the Chain Rule Calculator

Users should check these errors mentioned below to get accurate or better results.

  • Your expression of the function that you would like to differentiate should be placed in the input box.
  • Make sure that the function depends on the differentiation variable you specified.
  • Make sure the expression you are differentiating is written in the correct structure so that there is no red error found while you use the chain rule derivative calculator.

For access to our full suite of calculators, including the Derivative Limit Calculator and others, visit our All Calculators page. Explore various mathematical tools to aid in your problem-solving endeavors.

Frequently Asked Question

What is the Chain Rule

In calculus, the chain rule is a fundamental theorem that is used to calculate the derivative of a composite function. In simple words, the chain rule helps to find the rate of change of one variable with respect to another variable when the variables are linked through a chain of functions.

$$ \frac{d}{dx} \biggr( f(g(x)) \biggr) \;=\; f’ \biggr( g(x)) . g’(x) \biggr) $$

This expression means that if there is a function that is composed of another function then the derivative of the composite function with respect to x is the derivative of the function evaluated at g(x) and multiplied by the derivative of g.

How to Do Chain Rule

Some simple steps are given to let you know how to do chain rule,

  • Begin by identifying the composite function which means finding the inner function and the outer function containing the inner function
  • Calculate the derivative of the outer function with respect to the inner function
  • Calculate the derivative of the inner function with respect to the original function
  • Now combine the inner and outer functions by multiplying the derivative of the inner and outer functions.

How would you Use the Chain Rule to Find dz/ dt

If you are finding dz/ dt then it means you are dealing with the function which is a composite of two or more functions. However, to find dz/ dt follow the below steps,

  • Start with the expression for z in terms of x and y.
  • Write the expression for x and y and express them as a function of t
  • Now differentiate using the chain rule formula:

$$ \frac{dz}{dt} \;=\; \frac{\partial z}{\partial x} . \frac{dx}{dt} + \frac{\partial z}{\partial y} . \frac{dy}{dt} $$

  • Substitute the expressions x, y, and z and find the 𝜕z/𝜕x and 𝜕z/𝜕y with the values of x and y. Also, find out the partial derivatives dx/dt and dy/dt using the same expressions.
  • Enter the above values into the chain rule formula and calculate.

What is the Limit Chain Rule

In calculus, the limit chain rule is not a standard term but it refers to the application of the chain rule in the context of limits. This concept is used when there is a composite function and you want to calculate the limit of the function as the variables approach a certain value.

To apply this concept you should identify the composite function and set it as an inner function then calculate the limit when it approaches a certain value. After that, replace it with a new variable and calculate the limit of the function. Lastly, calculate the final limit.

Is there a Chain Rule for Integration

Yes, there is a chain rule for integration which is also called the integration of substitution or as the chain in variables rule, used to integrate composite functions. This change in variables rule or integration by substitution rule is equal to differentiation and used in simplifying the integrals by the substitutions.

For the chain rule of integration, first of all, identify the portion that you want to integrand which is the function whose derivative is present in the integrand. Now substitute the identified part and integrate the function. Substitute again with the original expression to get the integral in terms of the original function.