Solve the following derivative through its definition

Question

We have a function $$f(x) = \frac{5}{\sqrt{x} + 1}$$

and its definition states that $$f'(x) = \lim_{x \to 0}\frac{f(x+h)-f(x)}{h}.$$

Therefore, I attempted it by computing the following $$\lim_{x \to 0}\frac{\frac{5}{\sqrt{x+h} + 1} - \frac{5}{\sqrt{x} +1}}{h}.$$

Then I tried to find the common denominator,

$$\lim_{x \to 0} \frac{\frac{5(\sqrt{x} +1)-5(\sqrt{x+h} +1)}{(\sqrt{x} +1)(\sqrt{x + h} +1)}}{h} $$

and now I'm having trouble simplifying it.

in the limit $h\to 0$ not $x\to 0$. To find the general way, prove it for $f(x)=\frac {g(x)}{w(x)}. It's easier. — Haha, Oct 11 '14 at 01:14

score 0 · Answer 1 · answered Oct 11 '14 at 00:35

0

At the step where you have $$ \lim_{x \rightarrow 0} \frac{\frac{5}{(x+h)^{0.5}+1}-\frac{5}{x^{0.5}+1}}{h}, $$ I would rationalize the denominator of each fraction in the numerator before proceeding further: $$ \lim_{x \rightarrow 0} \frac{\frac{5((x+h)^{0.5}-1)}{((x+h)^{0.5}+1)((x+h)^{0.5}-1)}-\frac{5(x^{0.5}-1)}{(x^{0.5}+1)(x^{0.5}-1)}}{h} $$ Then the denominators will become simple, at least.

answered Oct 11 '14 at 00:35

Greg Martin

78,820

I'm still having trouble simplifying it... – KOFFEE Oct 11 '14 at 02:25
I agree that it's troublesome. However, at this point you could show us some effort and progress, so we can help you get past where you're stuck. – Greg Martin Oct 11 '14 at 17:28

Solve the following derivative through its definition

1 Answers1