Question

[1/(√x+h)-1/(√x)]/h

Answer 1

both the x and h are squared rooted in the first half

Answer 2

First get a common denominator for the numerator term: \[\begin{aligned}\dfrac{\dfrac{1}{\sqrt{x+h}}-\dfrac{1}{\sqrt{x}}}{h} &= \dfrac{\dfrac{\sqrt{x}}{\sqrt{x}\sqrt{x+h}} - \dfrac{\sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}}{h}\\ &= \dfrac{\dfrac{\sqrt{x}-\sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}}{h}\\ &= \dfrac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}}\end{aligned}\]The next step is to now multiply the numerator and the denominator by the conjugate of the numerator. Thus, we see that
\[\begin{aligned}\dfrac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}} &= \dfrac{\sqrt{x}-\sqrt{x+h}}{\color{red}h \sqrt{x}\sqrt{x+h}}\cdot\frac{\sqrt{x}+\sqrt{x+h}}{\sqrt{x}+\sqrt{x+h}}\\ &=\ldots\end{aligned}\]You want to continue simplifying this until the lone \(\color{red} h\) term in the denominator cancels out.

Can you take things from here? :-)

Answer 3

no sorry im still having trouble factoring the h out

Answer 4

What do you get when you simplify \((\sqrt{x}-\sqrt{x+h})(\sqrt{x}+\sqrt{x+h})\)? This part is the key to the entire simplification process.

Answer 5

Note that this looks like (a-b)(a+b). How does this product expand? :-)

Answer 6

-h

Answer 7

a²-b²

Answer 8

x-(√x+h)²

Answer 9

Correct. So you get \(-h\) on top, and hence the difference quotient simplifies to
\[\large \dfrac{-\color{red}h}{\color{red}h\sqrt{x}\sqrt{x+h}(\sqrt{x}+\sqrt{x+h})} = -\frac{1}{\sqrt{x}\sqrt{x+h}(\sqrt{x}+\sqrt{x+h})}\]In my opinion, this would be a good place to stop simplifying...but if they insist, you can then multiply out the denominator to get \(\large -\dfrac{1}{x\sqrt{x+h}+(x+h)\sqrt{x} }\).

Does this who process make more sense now? :-)

Answer 10

Thank you and it makes sense, but i will still ask my teacher for more help on these kinds of problems