chain rule


In calculus, the chain rule is a formula for finding the derivative of function that is the composition of two or more functions (see function composition). The chain rule says that if we have a function h composed of differentiable functions g and f such that h(x) = g(f(x)), then the derivative of h is calculated thusly:

    \[h'(x) = g'(f(x)) \cdot f'(x)\]

For the above example where h is the composition of two functions, the chain rule tells us to first take the derivative of the “outer” function g, leaving the “inner” function f alone, and multiply that result by the derivative of the “inner” function f.

If we assign f(x) to the variable y, such that y = f(x) and assign h(x) to the variable z, such that z = g(y), then z becomes a function of x by way of y, and we can write the chain rule in Leibniz’s notation:

    \[\dfrac{dz}{dx} = \dfrac{dz}{dy} \cdot \dfrac{dy}{dx}\]

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