Question

If P(x) =√x, show that P(x+h) - P(x) = h(√x+h + √x).

Answer 1

$P(x)=\sqrt{x},$

then

$P(x+h)=\sqrt{x+h}.$

Let's subtract and simplify:

$P(x+h) - P(x)=\sqrt{x+h}-\sqrt{x}=$

$= \frac{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}{\sqrt{x+h}+\sqrt{x}}=$

$= \frac{(\sqrt{x+h})^2-(\sqrt{x})^2}{\sqrt{x+h}+\sqrt{x}}= \frac{(x+h)-x}{\sqrt{x+h}+\sqrt{x}}= \frac{h}{\sqrt{x+h}+\sqrt{x}}.$

So, we showed that

$P(x+h) - P(x)= \frac{h}{\sqrt{x+h}+\sqrt{x}}.$