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Chain Rule

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Tap a term to see what it means, or .

Calculate

Example: the derivative of sin(x²) at x = 1. Pick the outer function, the inner one and the point to work out your own case.

Step by step

    Worked examples

    Sine of x squared

    1. The inner function is x², and the outer one, the sine:
    2. The derivative of the sine is the cosine, evaluated at the inner function; that of x² is 2x:

    Exponential of a line

    1. The derivative of eᵘ is itself, and that of 3x + 1 is 3:

    Two layers: the rule applied twice

    1. From the outside in: a cube, a logarithm and x² + 1.
    2. Differentiate the cube, then the logarithm, then the polynomial, and multiply everything:

    In statistics, the chain rule is behind the derivative of the normal density, e raised to −(x − μ)²/(2σ²), and behind every maximum-likelihood fit.

    What each term means

    Derivative of the Composite
    The rate of change of f(g(x)): one function applied to the result of another, such as sin(x²) or e^(3x + 1).
    Outer Derivative (f′(g(x)))
    Differentiate the outer function as if the inner one were a single variable, then put g(x) back in its place.
    Inner Derivative (g′(x))
    The correction for the inner function's speed: if g changes fast, the composite changes proportionally faster.