To find the derivative of the function [tex]\( f(x) = \ln(x^2 - 8x + 17) \)[/tex], we can use the chain rule of differentiation. The chain rule states that if you have a composite function [tex]\( g(h(x)) \)[/tex], then its derivative is [tex]\( g'(h(x)) \cdot h'(x) \)[/tex].
In this problem, [tex]\( g(u) = \ln(u) \)[/tex] and [tex]\( h(x) = x^2 - 8x + 17 \)[/tex].
Let's break this down step by step:
1. Identify the outer and inner functions:
- Outer function: [tex]\( g(u) = \ln(u) \)[/tex]
- Inner function: [tex]\( u = h(x) = x^2 - 8x + 17 \)[/tex]
2. Differentiate the outer function [tex]\( g(u) \)[/tex]:
- The derivative of [tex]\( \ln(u) \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( \frac{1}{u} \)[/tex].
3. Differentiate the inner function [tex]\( h(x) \)[/tex]:
- The derivative of [tex]\( x^2 - 8x + 17 \)[/tex] with respect to [tex]\( x \)[/tex] is obtained by applying the power rule and combining terms:
[tex]\[
\frac{d}{dx}(x^2 - 8x + 17) = 2x - 8
\][/tex]
4. Apply the chain rule:
- According to the chain rule, the derivative of [tex]\( f(x) = \ln(h(x)) = \ln(x^2 - 8x + 17) \)[/tex] is:
[tex]\[
f'(x) = \frac{d}{dx}(\ln(h(x))) = \frac{1}{h(x)} \cdot h'(x)
\][/tex]
- Substitute [tex]\( h(x) = x^2 - 8x + 17 \)[/tex] and [tex]\( h'(x) = 2x - 8 \)[/tex]:
[tex]\[
f'(x) = \frac{1}{x^2 - 8x + 17} \cdot (2x - 8)
\][/tex]
5. Simplify:
- Combine the expressions:
[tex]\[
f'(x) = \frac{2x - 8}{x^2 - 8x + 17}
\][/tex]
Thus, the derivative of the function [tex]\( f(x) = \ln(x^2 - 8x + 17) \)[/tex] is:
[tex]\[
f'(x) = \frac{2x - 8}{x^2 - 8x + 17}
\][/tex]
