Let's find the derivative of the function [tex]\( f(x) = \sqrt{x - 9} \)[/tex] step-by-step.
1. Rewrite the Function:
We start by expressing the square root in a more convenient form for differentiation:
[tex]\[
f(x) = \sqrt{x - 9} = (x - 9)^{1/2}
\][/tex]
2. Differentiate Using the Chain Rule:
To differentiate [tex]\( f(x) = (x - 9)^{1/2} \)[/tex], we apply the chain rule. The chain rule states that the derivative of a composite function [tex]\( g(h(x)) \)[/tex] is [tex]\( g'(h(x)) \cdot h'(x) \)[/tex]. Here, [tex]\( g(u) = u^{1/2} \)[/tex] and [tex]\( h(x) = x - 9 \)[/tex].
First, we differentiate the outer function with respect to [tex]\( u \)[/tex]:
[tex]\[
\frac{d}{du} \left( u^{1/2} \right) = \frac{1}{2} u^{-1/2} = \frac{1}{2} \cdot \frac{1}{\sqrt{u}}
\][/tex]
Next, we substitute [tex]\( u = x - 9 \)[/tex]:
[tex]\[
\frac{d}{du} \left( (x - 9)^{1/2} \right) = \frac{1}{2} \cdot \frac{1}{\sqrt{x - 9}}
\][/tex]
3. Differentiate the Inner Function:
The inner function [tex]\( h(x) = x - 9 \)[/tex] is straightforward to differentiate:
[tex]\[
\frac{d}{dx} (x - 9) = 1
\][/tex]
4. Combine the Results:
Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function:
[tex]\[
\frac{d}{dx} \left( (x - 9)^{1/2} \right) = \frac{1}{2} \cdot \frac{1}{\sqrt{x - 9}} \cdot 1 = \frac{1}{2\sqrt{x - 9}}
\][/tex]
Thus, the derivative of the function [tex]\( f(x) = \sqrt{x - 9} \)[/tex] is:
[tex]\[
f'(x) = \frac{1}{2\sqrt{x - 9}}
\][/tex]
