MathematicsCalculusA-Level
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Chain Rule Calculator

Differentiating composite functions.

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Total Derivative

Formula first

Overview

The Chain Rule is a fundamental formula in calculus used to find the derivative of a composite function. It establishes that the derivative of a nested function is the product of the derivative of the outer function and the derivative of the inner function.

Symbols

Variables

\frac{dy}{dx} = Total Derivative, \frac{dy}{du} = Outer Derivative, \frac{du}{dx} = Inner Derivative

Total Derivative
Outer Derivative
Inner Derivative

Apply it well

When To Use

When to use: Apply this rule when you need to differentiate a function that is composed of other functions, often described as a function within a function. It is necessary for expressions involving powers of polynomials, trigonometric functions with complex arguments, or exponential functions where the exponent is another function.

Why it matters: This rule is the foundation for many advanced mathematical concepts, including the backpropagation algorithm used to train neural networks in artificial intelligence. In physics and engineering, it allows for the analysis of related rates, such as how the volume of a sphere changes over time as its radius expands.

Avoid these traps

Common Mistakes

  • Forgetting the inner derivative.
  • Adding instead of multiplying.

One free problem

Practice Problem

In a calculus exercise involving a composite function, you determine that the derivative of the outer function with respect to its internal variable is 5, and the derivative of that internal variable with respect to x is 4. Calculate the total derivative dy/dx.

Outer Derivative5
Inner Derivative4

Solve for:

Hint: The Chain Rule states that the total derivative is the product of the outer and inner derivatives.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Chain rule
  2. Calculus (8th ed.) by James Stewart
  3. Stewart, James. Calculus: Early Transcendentals. Cengage Learning.
  4. Thomas, George B., et al. Thomas' Calculus. Pearson Education.
  5. Thomas' Calculus, 14th Edition, George B. Thomas, Maurice D. Weir, Joel Hass
  6. Calculus, 8th Edition, James Stewart
  7. AQA A-Level Mathematics — Pure (Differentiation)