Certainly! To find [tex]\(\frac{d}{d x}[f(g(x))]\)[/tex] using the chain rule, we follow these steps:
1. Identify the Composite Function:
We have a composite function [tex]\( f(g(x)) \)[/tex], which means [tex]\( f \)[/tex] is a function of [tex]\( g \)[/tex], and [tex]\( g \)[/tex] is a function of [tex]\( x \)[/tex].
2. Recall the Chain Rule Formula:
The chain rule for differentiation states that if you have a composite function [tex]\( (f \circ g)(x) \)[/tex], its derivative is given by:
[tex]\[
(f \circ g)'(x) = f'(g(x)) \cdot g'(x)
\][/tex]
3. Apply the Chain Rule:
Here,
- [tex]\( f'(g(x)) \)[/tex] is the derivative of the outer function [tex]\( f \)[/tex] evaluated at the inner function [tex]\( g(x) \)[/tex].
- [tex]\( g'(x) \)[/tex] is the derivative of the inner function [tex]\( g(x) \)[/tex] with respect to [tex]\( x \)[/tex].
4. Combine the Results:
Putting it all together, the derivative of the composite function [tex]\( f(g(x)) \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{d}{d x}[f(g(x))] = f'(g(x)) \cdot g'(x)
\][/tex]
Thus, using the chain rule, we obtain:
[tex]\[
\frac{d}{d x}[f(g(x))] = f'(g(x)) \cdot g'(x)
\][/tex]
This gives us the desired derivative of the composite function.
