To solve for the derivative of the function [tex]\( f(x) = (x^2 + 4x + 8)^4 \)[/tex] and then evaluate it at [tex]\( x = 5 \)[/tex], we can follow these steps:
### Finding [tex]\( f'(x) \)[/tex]:
1. Function Definition:
[tex]\( f(x) = (x^2 + 4x + 8)^4 \)[/tex]
To find [tex]\( f'(x) \)[/tex], we need to apply the chain rule. The chain rule for differentiation states that if you have a composite function [tex]\( g(h(x)) \)[/tex], then [tex]\( g'(h(x)) \cdot h'(x) \)[/tex].
2. Outer Function and Inner Function:
Here, the outer function is [tex]\( u^4 \)[/tex] where [tex]\( u = x^2 + 4x + 8 \)[/tex].
The inner function is [tex]\( x^2 + 4x + 8 \)[/tex].
3. Derivative of the Outer Function:
The derivative of [tex]\( u^4 \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( 4u^3 \)[/tex].
4. Derivative of the Inner Function:
The derivative of [tex]\( x^2 + 4x + 8 \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( 2x + 4 \)[/tex].
5. Apply Chain Rule:
By the chain rule, [tex]\( f'(x) = 4(u)^3 \cdot (2x + 4) \)[/tex].
6. Substitute the Inner Function:
So, [tex]\( f'(x) = 4(x^2 + 4x + 8)^3 \cdot (2x + 4) \)[/tex].
Simplify:
[tex]\[
f'(x) = (8x + 16) \cdot (x^2 + 4x + 8)^3
\][/tex]
So, the first part of the solution is:
[tex]\[
f'(x) = (8x + 16) \cdot (x^2 + 4x + 8)^3
\][/tex]
### Evaluating [tex]\( f'(5) \)[/tex]:
1. Substitute [tex]\( x = 5 \)[/tex] into the Derived Function:
[tex]\[
f'(5) = (8(5) + 16) \cdot (5^2 + 4(5) + 8)^3
\][/tex]
2. Simplify Inside the Parentheses:
[tex]\[
8(5) + 16 = 40 + 16 = 56
\][/tex]
[tex]\[
5^2 + 4(5) + 8 = 25 + 20 + 8 = 53
\][/tex]
3. Evaluate the Expression:
[tex]\[
f'(5) = 56 \cdot 53^3
\][/tex]
From this, we see that:
[tex]\[
f'(5) = 8,337,112
\][/tex]
Thus, the complete solution is:
[tex]\[
\begin{aligned}
f'(x) &= (8x + 16) \cdot (x^2 + 4x + 8)^3, \\
f'(5) &= 8,337,112.
\end{aligned}
\][/tex]
