Question

Find\[f'(x)\]for\[f(x)=\frac{1}{\sqrt{x+2}}\]using the definition of the derivative.

Answer 1

Someone gave me this problem last night and I couldn't get it. I must just be missing something simple I bet.

Answer 2

I would suggest you to do it again, with calm mind.

I think definition is, \[\lim_{h\to0} \frac{f(x+h) - f(x)}{h}\]

All you have to do is rationalization.

Answer 3

\[\large\lim_{h\to 0}\frac{\frac{1}{\sqrt{x+h+2}} - \frac{1}{\sqrt{x+2}} }{h} \times \frac{\frac{1}{\sqrt{x+h+2}}+ \frac{1}{\sqrt{x+2}}}{\frac{1}{\sqrt{x+h+2}}+ \frac{1}{\sqrt{x+2}}}\]

Answer 4

Rationalize the expression before using the definition.

Answer 5

that's exactly what I did, but I think somehow the fractions are confusing me. I can't seem to get rig of the h in the denominator. I know to rationalize in these cases, but this is just giving me a pain somehow.

Answer 6

after rationalizing take the lcm and add... make sure while adding u dont do any sign mistake... after doing this u will find that u can get rid of the h in the denominator

Answer 7

Okay, I am gonna leave the Denominator and Solve only the numerator.

\[\left(\frac{1}{\sqrt{x+h+2}} - \frac{1}{\sqrt{x+2}}\right) \times \left(\frac{1}{\sqrt{x+h+2}} + \frac{1}{\sqrt{x+2}}\right)\]\[\implies \frac{1}{x+2+h} - \frac{1}{x+2}\]
\[\frac{x+2 - x - 2-h}{(x+h+2)(x+2)} \implies \frac{-h}{(x+2+h)(x+2)}\]

Answer 8

ya exactly ishaan

Answer 9

gotchya

Answer 10

Yeah I just got confused... thanks for everybody's patience :)

Answer 11

\[\left(\frac{\sqrt{x+2} }{\sqrt{x+h+2}} \frac{-\sqrt{x+h+2}}{\sqrt{x+2}}\right) \times \left(\frac{\sqrt{x+2}}{\sqrt{x+h+2}} \frac{+\sqrt{x+h+2}}{\sqrt{x+2}}\right)\]