A Comprehensive Guide: Finding the Derivative of a Composite Function using the Chain Rule

d/dx f(g(x))

To find the derivative of the composite function f(g(x)) with respect to x, we can use the chain rule

To find the derivative of the composite function f(g(x)) with respect to x, we can use the chain rule. The chain rule states that if y = f(u) and u = g(x), then the derivative of y with respect to x is given by:

dy/dx = dy/du * du/dx.

In this case, let’s assume that f(u) is differentiable and g(x) is also differentiable.

So, we have:

y = f(g(x)).

Now, let’s find the derivative dy/dx:

dy/dx = df(g(x))/dx.

To simplify this, we’ll apply the chain rule:

dy/dx = df(g(x))/dg(x) * dg(x)/dx.

This gives us:

dy/dx = f'(g(x)) * g'(x).

So, the derivative of f(g(x)) with respect to x is:

d/dx f(g(x)) = f'(g(x)) * g'(x).

Note that f'(g(x)) represents the derivative of f with respect to g(x), while g'(x) represents the derivative of g with respect to x.

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