Integral Chain Rule Calculator

What is the Chain Rule?

The chain rule is a fundamental tool in calculus for finding the derivatives of composite functions. A composite function is a function that is composed of two or more functions. For example, if you have a function y = f(g(x)), then the derivative dy/dx can be found using the chain rule.

The chain rule extends to functions with more than two compositions. For example, if y = f(g(h(x))), then dy/dx = f'(g(h(x))) * g'(h(x)) * h'(x).

Integral Chain Rule

The integral chain rule is used to find the antiderivative of a composite function. It's essentially the reverse process of the chain rule for differentiation. The integral chain rule states that if you have an integral of a composite function, you can change the variable of integration to simplify the integral.

This technique is particularly useful when dealing with integrals that involve trigonometric, exponential, or logarithmic functions composed with other functions.

How to Use the Calculator

Our integral chain rule calculator makes it easy to compute derivatives of composite functions. Here's how to use it:

1. Enter the outer function in the first input field.
2. Enter the inner function in the second input field.
3. Click the "Calculate" button to see the result.
4. The calculator will display the derivative of the composite function using the chain rule.

Examples

Let's look at some examples to see how the integral chain rule works in practice.

Example 1: Simple Composition

Find the derivative of y = sin(3x).

Using the chain rule:

dy/dx = cos(3x) * 3 = 3cos(3x)

Example 2: Nested Composition

Find the derivative of y = e^(2x^2).

Using the chain rule:

dy/dx = e^(2x^2) * 4x = 4x e^(2x^2)

Example 3: Trigonometric Composition

Find the derivative of y = tan(5x).

Using the chain rule:

dy/dx = sec²(5x) * 5 = 5sec²(5x)