Mastering The Chain Rule In Calculus: A Step-By-Step Guide To Derivatives Of Composite Functions

Chain Rule (Using ‘ )

Function f(g(x))Derivative f'(g(x))g'(x)

In calculus, the chain rule is a method of finding the derivative of a function composed of two or more functions. The chain rule is used to find the derivative of the composite function f(g(x)), where f(x) and g(x) are two differentiable functions. The chain rule can be used to compute the derivative of a function that has a lot of nested functions.

The chain rule is often denoted using the prime notation. If we have a function u and a function y that is a composite of u and x, denoted as y = f(u(x)), then the derivative of y with respect to x is given by:

dy/dx = dy/du * du/dx

Here, f(u) is the outer function, and u(x) is the inner function. The chain rule states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

To use the chain rule, we first identify the inner function and the outer function. We then take the derivative of the outer function, evaluated at the inner function, and multiply this by the derivative of the inner function. Finally, we substitute the inner function back into the expression to get the derivative of the composite function.

For example, let’s consider the function y = sin(2x). Here, the inner function is 2x, and the outer function is sin(x). We can apply the chain rule as follows:

dy/dx = cos(2x) * d/dx(2x)
dy/dx = cos(2x) * 2
dy/dx = 2cos(2x)

So, the derivative of y = sin(2x) with respect to x is 2cos(2x).

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