Let's solve the problem step-by-step to find the difference quotient for the given function [tex]\( f(x) = \sqrt{17x} \)[/tex].
1. Define the function and the increment:
[tex]\[
f(x) = \sqrt{17x}
\][/tex]
We need to compute [tex]\( f(x + h) \)[/tex]:
[tex]\[
f(x + h) = \sqrt{17(x + h)}
\][/tex]
2. Form the difference quotient:
[tex]\[
\frac{f(x + h) - f(x)}{h}
\][/tex]
Substituting in the expressions for [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[
\frac{\sqrt{17(x + h)} - \sqrt{17x}}{h}
\][/tex]
3. Simplify the expression:
First, note that both terms under the square roots have a common factor of 17. We can factor this out and simplify:
[tex]\[
\frac{\sqrt{17} \cdot \sqrt{x + h} - \sqrt{17} \cdot \sqrt{x}}{h}
\][/tex]
[tex]\[
\frac{\sqrt{17} (\sqrt{x + h} - \sqrt{x})}{h}
\][/tex]
Now, multiply the numerator and the denominator by the conjugate of the numerator to simplify:
[tex]\[
\frac{\sqrt{17} (\sqrt{x + h} - \sqrt{x})}{h} \cdot \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}}
\][/tex]
This step rationalizes the numerator:
[tex]\[
\frac{\sqrt{17} (\sqrt{x + h} - \sqrt{x})(\sqrt{x + h} + \sqrt{x})}{h (\sqrt{x + h} + \sqrt{x})}
\][/tex]
4. Simplify further:
The numerator is:
[tex]\[
(\sqrt{x + h} - \sqrt{x})(\sqrt{x + h} + \sqrt{x}) = (x + h) - x = h
\][/tex]
With this simplification, our difference quotient becomes:
[tex]\[
\frac{\sqrt{17} \cdot h}{h (\sqrt{x + h} + \sqrt{x})}
\][/tex]
The [tex]\( h \)[/tex] terms cancel out:
[tex]\[
\frac{\sqrt{17}}{\sqrt{x + h} + \sqrt{x}}
\][/tex]
Thus, the simplified difference quotient for the function [tex]\( f(x) = \sqrt{17x} \)[/tex] is:
[tex]\[
\boxed{\frac{\sqrt{17} (\sqrt{x + h} - \sqrt{x})}{h}}
\][/tex]
