To understand how to differentiate composite functions using the Chain Rule, we first need to know what a composite function is.

A composite function is made by combining two functions. We can write it as ( f(g(x)) ), where ( f ) is one function and ( g ) is another.

The Chain Rule helps us find the derivative (or the rate of change) of this composite function. Here’s how it works:

1. First, you take the derivative of the outer function.
2. Keep the inner function as it is.
3. Then, multiply that result by the derivative of the inner function.

Here’s the formula for the Chain Rule:

[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) ]

Now, let's look at an example. Imagine we have the function ( h(x) = (3x + 2)^4 ).

In this case:

• The outer function is ( f(u) = u^4 )
• The inner function is ( g(x) = 3x + 2 )

Now, let's go through the steps:

1. Differentiate the outer function:
   The derivative is ( f'(u) = 4u^3 ).
   So when we substitute back in, we have ( f'(g(x)) = 4(3x + 2)^3 ).

2. Differentiate the inner function:
   The derivative is ( g'(x) = 3 ).

Now, let’s put it all together:

[ h'(x) = f'(g(x)) \cdot g'(x) = 4(3x + 2)^3 \cdot 3 = 12(3x + 2)^3 ]

And that’s how you use the Chain Rule!