How to Find the Derivative of a Composite Function Using the Chain Rule

d/dx f(g(x))

To find the derivative of a composite function, such as f(g(x)), we can use the chain rule

To find the derivative of a composite function, such as f(g(x)), we can use the chain rule. The chain rule states that if we have a function f(u) where u is a function of x, and we want to find the derivative of f(u) with respect to x, we need to multiply the derivative of f(u) with respect to u by the derivative of u with respect to x.

In this case, we have f(g(x)), where g(x) is the inner function and f(u) is the outer function.

To find the derivative of f(g(x)), we follow these steps:

1. Compute the derivative of g(x) with respect to x. This gives us g'(x).

2. Replace u in f(u) with g(x). This gives us f(g(x)).

3. Compute the derivative of f(u) with respect to u. This gives us f'(u).

4. Multiply g'(x) and f'(u) together. This gives us the derivative of f(g(x)) with respect to x.

In mathematical notation, this can be written as:

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

Let’s work through an example to illustrate this:

Suppose we have f(u) = u^2 and g(x) = 3x + 2. We want to find the derivative of f(g(x)) with respect to x.

1. Compute g'(x):

g(x) = 3x + 2
g'(x) = d/dx [3x + 2]
= 3

2. Replace u in f(u) with g(x):

f(g(x)) = f(3x + 2)

3. Compute the derivative of f(u) with respect to u:

f(u) = u^2
f'(u) = d/du [u^2]
= 2u

4. Multiply g'(x) and f'(u):

f'(g(x)) = 2(3x + 2)

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

d/dx [f(g(x))] = 2(3x + 2)

Simplifying further:

d/dx [f(g(x))] = 6x + 4

More Answers: