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Differentiate a polynomial using the power rule.
Key values: f(x) = x^3 - 2x^2 + x · 1st derivative
Differentiate a composite function with sin(x^2).
Key values: f(x) = sin(x^2) · Chain rule applied
Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.
Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0
Compute the second derivative to analyze concavity.
Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative
Enter a mathematical expression and click Calculate to see the symbolic derivative with step-by-step solution and interactive graph.
The chain rule differentiates composite functions — functions nested inside other functions. If , then:
In words: differentiate the outer function (leaving the inner function unchanged), then multiply by the derivative of the inner function.
Leibniz notation: If and , then . The terms “cancel” — a useful mnemonic, though not rigorous.
| Function | Outer | Inner | Derivative |
|---|---|---|---|
When three or more functions are nested, the chain rule extends naturally. For :
Example:
The chain rule differentiates composite functions . The derivative is : differentiate the outer function leaving the inner unchanged, then multiply by the derivative of the inner function.
Use the chain rule whenever you see a function nested inside another function. For example, is applied to , and is the exponential applied to . If the argument is anything other than plain , the chain rule applies.
The most common mistake is forgetting to multiply by the inner derivative. For example, , not just . The factor of 3 from the inner function is essential.
For , the derivative is . Each layer contributes a factor. For example, .
The chain rule applies to composition (one function inside another), like . The product rule applies to multiplication (two functions multiplied together), like . Misidentifying composition as multiplication is a common error.