Chain Rule
The reason derivative of the composite function is called as "Chain Rule" is because the quantity "u" is an independent variable and dependent variable both and occurs in both the bottom of dy/du and numerator of du/dx. Keep in mind that dy/dx, dy/du, and du/dx are NOT fractions though and you cannot be tempted to divide "u". As a review, recall the meanings of the derivative symbols.
- dy/dx is derivative of y with respect to x, x is an independent variable.
- dy/du is derivative of y with respect to u, u is an independent variable here.
- du/dx is derivative of u with respect to x, is an independent variable.
If f and g are the 2 functions defined by y = f(u) and u = g(x) respectively then the function defined by y = f [g(x)] or fog(x) is known as composite function or a function of a function.
The theorem for finding the derivative of the composite function is called as CHAIN RULE.
Theorem:
If f and g are differentiable and are defined by y = f (u) and u = g (x) respectively, then the composite function y = f [ g (x) ] is differentiable and we have
dy/dx = dy/du * du/dx
Corollary:
If y = f (u), u = g (v) and v = h (x) where f, g and h are differentiable functions of u, v and x respectively, then we obtain the equation which is stated as follows
dy/dx = dy/du * du / dv * dv /dx
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