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.

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Introduction to Chain Rule Calculator:

Chain rule calculator is an online tool that is used to find the differentiation of two or more composite functions. It evaluates the complex function that cannot be solved by other differential methods directly.

Chain Rule Calculator with Steps

Chain rule derivative calculator is a beneficial tool for students, professionals and teachers to get the solution of complex chain rule differential function problems without any error.

What is Chain Rule?

Chain rule is a process of differentiation, 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 to identify if the given differential function is a chain rule problem or not. If you don't know about chain rule process then follow the given steps to evaluate chain rule.

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, add 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 to get the solution. Follow our guidelines before using it:

Enter the function f(x) of 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.
The “Calculate” button gives you the solution of your given chain rule differential problem.
If you want to try our tool for the first time then it is suggested to try the load example to check its working process.
The “Recalculate” button of derivative of composite function calculator brings back a new page for finding more example solutions of differential problems.

Output from Chain Rule Solver:

Chain rule calculator with steps gives you the solution of chain rule differential question when you give it an input. The results include:

Result Option:

When you click on the result option, it gives you the solution of chain rule problem.

Possible Steps:

It provides you step by step solution of chain rule differential problem.

Advantages of Composite Function Derivative Calculator:

This calculator provides you many benefits that you will help to calculate the derivative chain rule problems. These advantages are:

Chain rule solver is a free tool that enables you to evaluate the chain rule differential problems.
It is a manageable tool that can solve trigonometry, logarithmic, and exponential functions for getting solution of chain rule derivative problem.
Our chain rule derivative calculator helps you to get a strong hold on chain rule concept by giving you an opportunity to practice complex questions.
It saves the time that you consume in the calculation of chain rule differential problem.
Composite derivative calculator provides exact solutions as per your input value and 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,