Question: $$\frac{\frac{1}{\sqrt {x+h}}- \frac{1}{\sqrt x}}{h}$$
Solution given:
$$= \frac{1}{h} \cdot\frac{\sqrt x - \sqrt {x+h}} {\sqrt {x + h}\sqrt x} $$
$$= \frac{1}{h} \cdot\ \frac{x - (x+h)}{\sqrt{x + h} \sqrt x (\sqrt{x} + \sqrt{x + h})}$$
$$= \frac{1}{h} \cdot\ \frac{x - x - h}{x \sqrt{x + h} + (x + h) \sqrt x}$$
$$= \frac{1}{h} \cdot\ \frac{-h}{x \sqrt{x + h} + (x + h) \sqrt x}$$
$$= -\frac{1}{x \sqrt{x + h} + (x + h) \sqrt x}$$
I've studied and understood the material up to this point just fine. I get about rationalizing stuff, conjugate pairs etc, but I can't figure out of what the author has between each step to get to the next.
I can only comprehend the first and possibly the second step. Source: http://www.themathpage.com/alg/multiply-radicals.htm
See problem 10, the last problem on the page.
Step 1: Delete the chain break and multiply with the denominator of the other fraciton
Step 2: Use Binomial formula since $(a-b)(a+b)=a^2-b^2$
Step 3: Use Associative law of addition multiply everything out in the denominator
Step 4: $(x-x)=0$
Step 5: $\frac{-h}{h} = -1$