Understanding the Chain Rule and How to Differentiate Composite Functions

d/dx f(g(x))

To differentiate the composite function f(g(x)) with respect to x, we will use the chain rule

To differentiate the composite function f(g(x)) with respect to x, we will use the chain rule.

First, let’s define the functions involved:

– f(x) is a function of x.
– g(x) is a function of x.

Now, we want to differentiate the composite function:

d/dx f(g(x))

To find the derivative, we consider the chain rule, which states that the derivative of a composite function is given by:

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

Here, f'(x) represents the derivative of f(x), and g'(x) represents the derivative of g(x).

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

1. Find the derivative of f'(x). Let’s say f'(x) = h(x).
2. Find the derivative of g(x). Let’s say g'(x) = k(x).
3. Substitute g(x) and g'(x) into h(x) and k(x), respectively. This gives us f'(g(x)) and g'(x), which we will now represent as h(g(x)) and k(x).
4. Finally, combine h(g(x)) and k(x) to form the derivative of the composite function:

d/dx f(g(x)) = h(g(x)) * k(x)

In summary, to differentiate a composite function, we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function.

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