function --- derivative (help!)

**gomes** · May 22nd 2010, 12:01 PM

but instead of f(x)=x^-2, let f(x) = x^(1/2 )

So, I did
[ (x+h)^1/2 - (x)^1/2 ]/ h
But im not sure how to simplify it. What would be my next step? Thanks alot

**harish21** · May 22nd 2010, 12:26 PM

Originally Posted by gomes

but instead of f(x)=x^-2, let f(x) = x^(1/2 )

So, I did
[ (x+h)^1/2 - (x)^1/2 ]/ h
But im not sure how to simplify it. What would be my next step? Thanks alot

wait..

$f(x) = x^{-2} \implies f(x) = \frac{1}{x^2}$

now using the definition

$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

$\lim_{h \to 0} \frac{{\frac{1}{(x+h)^2}}-{\frac{1}{x^2}}}{h}$

simplify this and get your proof

**skeeter** · May 22nd 2010, 01:49 PM

Originally Posted by gomes

but instead of f(x)=x^-2, let f(x) = x^(1/2 )

$\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h}$

$\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}<br />$

$\lim_{h \to 0} \frac{(x+h) - x}{h(\sqrt{x+h} + \sqrt{x})}<br />$

$\lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}}$

finish it.

**gomes** · May 22nd 2010, 02:37 PM

Thanks everyone

Originally Posted by skeeter

$\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h}$

$\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}<br />$

$\lim_{h \to 0} \frac{(x+h) - x}{h(\sqrt{x+h} + \sqrt{x})}<br />$

$\lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}}$

finish it.

Thanks, I'm having trouble with sin(2x), i think my double angle formulae, i might be doing something wrong/missing something. Any help? Most appreciated!

**gomes** · May 22nd 2010, 02:47 PM

Sorry, one more, im stuck

**AllanCuz** · May 22nd 2010, 07:07 PM

Originally Posted by gomes

Sorry, one more, im stuck

I'm lazy so I wont reproduce this here but look at : http://www.mash.dept.shef.ac.uk/Reso...principles.pdf

If you have any questions we can answer them here

**gomes** · May 23rd 2010, 10:03 AM

Originally Posted by AllanCuz

I'm lazy so I wont reproduce this here but look at : http://www.mash.dept.shef.ac.uk/Reso...principles.pdf

If you have any questions we can answer them here

Thanks! i understand it, but what about for sin2x?

**harish21** · May 23rd 2010, 12:47 PM

Originally Posted by gomes

Thanks! i understand it, but what about for sin2x?

$\lim_{h \to 0} \frac{sin(2x+2h)-sin(2x)}{h}$

$= \lim_{h \to 0} \frac{2}{2} \times \frac{sin(2x+2h)-sin(2x)}{h}$

$= \lim_{h \to 0} \frac{2(sin(2x+2h)-sin(2x))}{2h}$

$= \lim_{h \to 0} \frac{2(sin(X+H)-sin(X))}{H}$ where X=2x and H=2h

$= 2cos(X)$ ; which you proved before doing this.

$= 2cos(2x)$

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