Chain Rule
f'(g(x))g'(x)
The chain rule is a derivative rule that is used to find the derivative of composite functions. It states that if we have a composite function f(g(x)), where g(x) is differentiable at x and f(x) is differentiable at g(x), then the derivative of the composite function is given by:
(f(g(x)))’ = f'(g(x)) * g'(x)
In other words, to find the derivative of a composite function, we first find the derivative of the outer function (f’) evaluated at the inner function (g(x)), and then multiply this by the derivative of the inner function (g'(x)).
Here’s an example to illustrate how the chain rule works:
Let’s say we have the function y = (cos(x^2))^3. To find the derivative of this function, we can use the chain rule as follows:
1. Let u = cos(x^2).
2. Then y = u^3.
3. Using the power rule, we can find du/dx = -2x * sin(x^2).
4. Using the chain rule, we have dy/dx = 3u^2 * du/dx.
5. Substituting u = cos(x^2) and du/dx = -2x * sin(x^2), we get:
dy/dx = 3(cos^2(x^2)) * (-2x*sin(x^2))
dy/dx = -6x(cos^2(x^2))*(sin(x^2))
Therefore, the derivative of y = (cos(x^2))^3 is dy/dx = -6x(cos^2(x^2))*(sin(x^2)).
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