The Chain Rule
What is the Chain Rule?
The chain rule is a formula used to calculate the derivative of a composite function. A composite function is a function that is formed by plugging one function into another function. For example, if we have the function f(x) = sin(x), and we plug the function g(x) = x^2 into f(x), we get the composite function h(x) = sin(x^2).
Chain Rule Formula
The chain rule formula is: ``` dy/dx = dy/du * du/dx ``` where: * dy/dx is the derivative of the composite function h(x) * dy/du is the derivative of the outer function f(u) * du/dx is the derivative of the inner function g(x)
Proof of the Chain Rule
The proof of the chain rule is a bit complex, but it can be found in most calculus textbooks.
Example
Let's use the chain rule to find the derivative of the function h(x) = sin(x^2).
``` dy/dx = dy/du * du/dx = cos(u) * 2x = cos(x^2) * 2x ```So, the derivative of h(x) is h'(x) = cos(x^2) * 2x.
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