Chain Rule
f(g(x)) = f'(g(x)) g'(x)
The chain rule is a calculus rule used to find the derivative of a composite function. A composite function is a function that is formed by combining two or more functions. The chain rule is used when we need to find the derivative of a composite function that cannot be easily found by using basic differentiation rules.
The chain rule is based on the following principle: if y is a function of u, and u is in turn a function of x, then the derivative of y with respect to x is given by the product of the derivative of y with respect to u and the derivative of u with respect to x.
Mathematically, if y = f(u) and u = g(x), then the derivative of y with respect to x is:
(dy/dx) = (dy/du) * (du/dx)
Here, (dy/du) represents the derivative of y with respect to u, and (du/dx) represents the derivative of u with respect to x.
To apply the chain rule, we need to first identify the function that is composed of two or more functions, and then differentiate each of the individual functions separately using the basic differentiation rules. Finally, we substitute these derivatives in the chain rule formula to get the derivative of the composite function.
The chain rule is an essential tool in calculus, and is used to solve problems in many areas of math and science, including physics, engineering, and economics.
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