Certainly! Let's find the derivative of the function [tex]\( f(x) = (x^2 + 3)^3 \)[/tex].
To find the derivative of a composite function, we can apply the chain rule. The chain rule states that if you have a composite function [tex]\( g(h(x)) \)[/tex], then its derivative is:
[tex]\[
(g(h(x)))' = g'(h(x)) \cdot h'(x)
\][/tex]
### Step-by-Step Solution:
1. Identify the outer and inner functions:
- The outer function [tex]\( g(u) \)[/tex] is [tex]\( u^3 \)[/tex].
- The inner function [tex]\( h(x) \)[/tex] is [tex]\( x^2 + 3 \)[/tex].
2. Differentiate the outer function [tex]\( g(u) \)[/tex] with respect to [tex]\( u \)[/tex]:
- The derivative of [tex]\( u^3 \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( 3u^2 \)[/tex].
3. Differentiate the inner function [tex]\( h(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
- The derivative of [tex]\( x^2 + 3 \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( 2x \)[/tex].
4. Apply the chain rule:
- According to the chain rule, the derivative of [tex]\( f(x) = (x^2 + 3)^3 \)[/tex] is:
[tex]\[
f'(x) = g'(h(x)) \cdot h'(x)
\][/tex]
5. Substitute [tex]\( h(x) \)[/tex] into [tex]\( g'(u) \)[/tex]:
- Since [tex]\( g'(u) = 3u^2 \)[/tex] and [tex]\( u = x^2 + 3 \)[/tex], we substitute [tex]\( u \)[/tex]:
[tex]\[
g'(h(x)) = 3(x^2 + 3)^2
\][/tex]
6. Multiply by the derivative of the inner function [tex]\( h'(x) \)[/tex]:
- We already found [tex]\( h'(x) = 2x \)[/tex].
- Thus, [tex]\( f'(x) \)[/tex] becomes:
[tex]\[
f'(x) = 3(x^2 + 3)^2 \cdot 2x
\][/tex]
7. Combine the terms:
- Simplify the expression:
[tex]\[
f'(x) = 6x (x^2 + 3)^2
\][/tex]
Therefore, the derivative of [tex]\( f(x) = (x^2 + 3)^3 \)[/tex] is:
[tex]\[
f'(x) = 6x (x^2 + 3)^2
\][/tex]
And this is the complete derivative of the function.
