Question

Evaluate the difference quotient and simplify the result.

f(x)= 1/√x

(f(x) - f(5)) / (x-5)

Answer 1

Hey c:

Answer 2

\[\Large\rm \frac{\color{orangered}{f(x)}-f(5)}{x-5}=\quad\frac{\color{orangered}{\frac{1}{\sqrt x}}-f(5)}{x-5}\]Understand what f(5) means?

Answer 3

yeah it's just 1/√5, right?

Answer 4

\[\frac{ 1 }{ \sqrt{5} }\] ***

Answer 5

Ok good.\[\Large\rm \frac{\frac{1}{\sqrt x}-\frac{1}{\sqrt 5}}{x-5}\]Mmm ok good.
So to simplify...

Answer 6

We want to start by turning this: \(\Large\rm \dfrac{1}{\sqrt x}-\dfrac{1}{\sqrt5}\) into a single fraction.

Answer 7

Remember how to find common denominator and all that jazz? :d

Answer 8

multiply by the conjugate?

Answer 9

Mmmmm I don't think that's the route we wanna go with this one.
The first denominator is missing a factor of sqrt(5),
while the second fraction is missing a factor of sqrt(x),\[\Large\rm \dfrac{1}{\sqrt x}-\dfrac{1}{\sqrt5}\quad=\quad \color{royalblue}{\frac{\sqrt5}{\sqrt5}}\cdot\dfrac{1}{\sqrt x}-\dfrac{1}{\sqrt5}\cdot\color{orangered}{\frac{\sqrt x}{\sqrt x}}\]\[\Large\rm \frac{\sqrt5}{\sqrt{5x}}-\frac{\sqrt x}{\sqrt{5x}}\quad=\quad \frac{\sqrt5-\sqrt x}{\sqrt{5x}}\]Understand what I was doing there? :d