chain rule

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The following rule that gives the derivative of the composition of two functions: If h(x)=(fg)(x)=f(g(x) for all x, then h′(x)=f′(g(x)g′(x). For example, if h(x)=(x2+1)3, then h=fg, where f(x)=x3 and g(x)=x2+1. Then f′(x)=3x2 and g′(x)=2x. So h′(x)=3(x2+1)22x=6x(x2+1)2. Another notation can be used: if y=f(g(x), write y=f(u), where u=g(x). Then the chain rule says that dy/dx=(dy/du)(du/dx). As an example of the use of this notation, suppose that y=(sin x)2. Then y=u2, where u=sin x. So dy/du=2u and du/dx=cos x, and hence dy/dx=2 sin x cos x.

Subjects: Mathematics.

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