Understanding the Chain Rule in Derivatives | A Comprehensive Explanation with Examples

d/dx[f(g(x))]

The expression d/dx[f(g(x))] represents the derivative of a composite function

The expression d/dx[f(g(x))] represents the derivative of a composite function. To find this derivative, we can apply the chain rule, which is a rule for finding the derivative of a composition of functions.

The chain rule states that if we have a composite function f(g(x)), where g(x) represents an inner function and f(u) represents an outer function, then the derivative of f(g(x)) with respect to x is given by:

d/dx[f(g(x))] = f'(g(x)) * g'(x)

Here, f'(u) represents the derivative of the outer function f(u) with respect to u, and g'(x) represents the derivative of the inner function g(x) with respect to x.

Therefore, to find the derivative of d/dx[f(g(x))], we need to compute the derivatives f'(g(x)) and g'(x), and then multiply them together.

It’s important to note that without knowing the specific functions f(u) and g(x), we cannot determine the exact value of d/dx[f(g(x))]. This can only be done when the functions f(u) and g(x) are given explicitly. However, we can express the derivative in terms of f'(u) and g'(x) using the chain rule.

To summarize:
d/dx[f(g(x))] = f'(g(x)) * g'(x)

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