Chain Rule Calculator

Want to calculate the chain rule to find the derivative of two or more composite functions? Try our chain rule calculator to evaluate the derivative of these functions.

with respect to

Feedback

How easy was it to use our calculator? Did you face any problem tell us!

Get the Widget

Add this powerfull tool to your website and help them balance chemical equations in second

Introduction To Chain Rule Calculator:

Chain rule Calculator is an online tool that is used to find the differentiation of two or more than two composite functions. Our chain rule derivative calculator evaluates the complex function which is made up of two or more than two functions which are not solved by other differential methods directly.

Chain Rule Calculator with Steps

It is a beneficial tool for students, professionals and teachers to get the solution of complex chain rule differential function problems in less than a minute without any error.

What Is Chain Rule?

Chain rule is a process of differentiation that is used to solve the composite function problems. This method is used when you have a complicated function that cannot be solved with other derivative rule methods (product rule,quotient rule, etc).

Suppose you have a composite function h(x) that can be expressed as h(x)=f(g(x))h(x) , where f and g are the functions of x, then the chain rule for derivative function h(x) is.

$$ h’(x) \;=\; f’(g(x)) . g’(x) $$

How To Calculate Chain Rule?

To calculate the chain rule you need to know the basic method of derivation so that you can easily identify if the given differential function is a chain rule problem or not.If you don't know about chain rule process then let us evaluate the process of chain rule problem of differentiation in steps.

Step 1:

Identify the Chain rule function which should be a composite function

Step 2:

Determine the inner function f(x) and the outer function g(x) for chain rule derivation.

Step 3:

Find the derivative of the outer function g`(x), leave the inner function f(x) as constant.

Step 4:

Find the derivative of the inner function f`(x) in chain rule differentiation.

Step 5:

Then adding the result of derivation of the inner function f`(x) and the outer function g`(x) in the chain rule formula.

$$ h’(x) \;=\; f’(g(x)) . g’(x) $$

Step 6:

Simplify the chain rule derivative function after putting the function value in it.

Solved Example Of Chain Rule:

The solved example of chain rule problem will help you to understand the method of finding chain rule differential problem.

Example: find the derivative of the following:

$$ h(x) \;=\; \frac{1}{(3x^2 + 1)^2} $$

Solution:

Identify the h(x) and g(x) function.

$$ h(x) \;=\; \frac{1}{(3x^2 + 1)^2} \;=\; (3x^2 + 1)^{-2} $$

$$ g(x) \;=\; 3x^2 + 1 $$

Differentiate the function h(x) and g(x),

$$ h’(x) \;=\; -2(3x^2 + 1)^{-3} $$

$$ g`(x)= 6x $$

Put the h(x) and g(x) in the chain rule formula,

$$ h’(x) \;=\; -2(3x^2 + 1)^{-3} 6x $$

Simplify it,

$$ h’(x) \;=\; \frac{-12x}{(3x^2 + 1)^3} $$

How To Use Chain Rule Derivative Calculator?

The composite function derivative calculator has a user-friendly interface, you just need to put a differential chain question in this calculator to get a solution without doing anything else. Follow our guidelines before using it. These guidelines are:

Enter the function f(x) of the chain rule that you want to differentiate in the input box.
Select the variable of differentiation in the input box.
Check the given derivative value before hitting the calculate button to start the evaluation process in the calculator.
The “Calculate” button gives you the solution of your given chain rule differential problem.
If you want to try out our calculator for the first time then it must be suggested to try out the load example to check its working process.
The “Recalculate” button of the derivative of composite function calculator brings back a new page for finding more example solutions of differential problems.

Output Of Chain Rule Solver:

Chain rule calculator with steps gives you the solution to a given chain rule differential question when you add the input into it. It may contain as:

Result Option:

When you click on the result option it gives you a solution to the chain rule problem.

Possible Steps:

It provides you with a solution to the chain rule differential problem where calculation steps are included.

Useful Features Of Composite Function Derivative Calculator:

This calculator provides you with many useful features that you get when you calculate the derivative chain rule problems and provides solutions. These features are:

Chain rule solver is a free tool that enables you to evaluate the chain rule differential problems freely.
It is a manageable tool that can solve different types of functions like logarithmic, exponential, and trigonometric to find the solution of chain rule derivative problem.
Our Chain rule derivative calculator helps you to get a stronghold on the chain rule concept when you use it for practice.
It saves the time that you consume on the calculation of the chain rule differential problem solution in a few seconds.
Composite derivative calculator provides exact solutions as per your input value when you use it to calculate the chain rule differential problems without any error.
The Derivative of composite function calculator is an educational tool that helps you to teach your children, and students in a simple and fun way.

Related References

How would you define chain rule?
Explain the chain rule in detail:
Give some Examples of chain rule:
Give an overview of chain rule?

Frequently Ask Questions

Solve the chain rule derivative examples y=ln( 3t+ 5t^4+ 7)

$$ y \;=\; ln(3t + 5t^4 + 7) $$

$$ g(t) \;=\; 3t + 5t^4 + 7 $$

Here, y is a composite function where f(u)=ln⁡(u) or u=g(t)and g(t)=5t^4+3t+7. Compute the derivative of y with respect to u:

$$ f’(u) \;=\; \frac{1}{u} $$

Substituting u = 3t + 5t^4 + 7, we get,

$$ f’ (g(t)) \;=\; \frac{1}{3t} + 5t^4 + 7 $$

Take the derivative of u with respect to t:
$$ \frac{du}{dt} \;=\; 20t^3 + 3 $$

Combine using the chain rule:
$$ \frac{dy}{dt} \;=\; \frac{dy}{du} ⋅ \frac{du}{dt} $$

$$ \frac{dy}{dt} \;=\; f’(g(t)) . g’(t) \;=\; \frac{1}{3t} + 5t^4 + 7 . (3 + 20t^3) $$

$$ \frac{dy}{dt} \;=\; \frac{3 + 20t^4}{3t + 5t^4 + 7} $$

Solve the chain rule derivative examples with cos(rsink(r)

$$ f(r) \;=\; cos(r sin(r) $$

Differentiating f(r)=cos⁡(r sin⁡(r))with respect to r. Use chain rule for differentiation and identify the outer function and inner function:

Outer function: cos⁡(x).

Inner function: x = r sin⁡(r)

Differentiate the outer function:

$$ f`(x) \;=\; cosx \;=\; −sin⁡(x) $$

Differentiate the inner function x=rsin⁡(r)x = r

Use the product rule
$$ \frac{d}{dr}[r sin⁡(r)] \;=\; sin⁡(r) + rcos⁡(r) $$

Apply the chain rule:
$$ k′(r) \;=\; −sin⁡(r sin⁡(r)) ⋅ \frac{d}{dr}r\; sin⁡(r) $$

Substituting the derivative of the inner function:
$$ k′(r) \;=\; −sin⁡(r sin⁡(r)) ⋅ (sin⁡(r) + rcos⁡(r) $$

Why do we calculate the multivariable chain rule

The multivariable chain rule is an important rule in calculus that uses the concept of the chain rule from single-variable functions to functions of several variables. It gives a better understanding of how a function changes in one or more variables that affect a composite function. It is used in the machine learning, statistics, and engineering fields.

By applying the multivariable chain rule, we can break down complex dependencies into small parts, allowing for more effective analysis and problem-solving techniques for multi-variable problems.

Solve the second derivative chain rule examples sin and cosine

For the differentiation of the sine or cosine chain rule, take an example with a solution.

Example:

Second derivative of, $$ h(x) \;=\; sin(x)\; cos(x) $$

Solution:

Differential with respect to x

$$ h’(x) \;=\; \frac{d}{dx} sin(x) cos(x) $$

$$ h’(x) \;=\; cos(x) . \frac{d}{dx} [sin(x)] + sin(x) . \frac{d}{dx} [cos(x)] $$

$$ h’(x) \;=\; cos(x) . cos(x) + sin(x) . (-sin(x)) $$

$$ h’(x) \;=\; cos^2(x) - sin^2(x) $$

Differentiate again with respect to x,

$$ h’’(x) \;=\; \frac{d}{dx} [cos^2 (x) - sin^2(x)] $$

$$ h’’(x) \;=\; 2cos(x) . (-sin(x)) - 2sin(x) . cos(x) $$

$$ h’’(x) \;=\; -2cos(x) sin(x) - 2cos(x) sin(x) $$

$$ h’’(x) \;=\; -4cos(x) sin(x) $$

$$ h(x) \;=\; sin(x)\; cos(x) ⇒ h’’(x) \;=\; -4cos(x)\; sin(x) $$

How to find tangent line chain rule

To Find the tangent line to a curve let's use the chain rule to differentiate composite functions, which is essential for determining the slope of the tangent line to a curve.

Examples:

Tangent line to, $$ y \;=\; sin(x^2)\; at\; x \;=\; 1 $$

Solution:

$$ f’(x) \;=\; \frac{d}{dx}[sin(x^2)] \;=\; cos(x^2) . \frac{d}{dx}[x^2] \;=\; cos(x^2) . 2x $$

Tangent at x = 1

$$ f’(x) \;=\; cos(1^2) . 2 . 1 \;=\; 2cos(1) $$

Equation of the Tangent Line:

$$ y − y_0 \;=\; m(x − x_0) $$

Where $$ m \;=\; \frac{dy}{dx}\; at\; x_0 $$

Put the value in the above equation,

$$ y - sin(1) \;=\; 2cos(1)(x - 1) $$

The tangent line to y = sin(x^2) at x = 1 is,