Help finding the derivative of rational/radical function

[jmanna98]

Please help me break down the first couple parts of this derivative question. This question gets a bit ugly:

Find the derivative of F(x)=(-1/sqrt(2x)) +2x

 

[SteamKing]

You can rewrite F(x) = 2x - (2x)^(-1/2)

Try taking the derivative of this expression.

 

[jmanna98]

ya but then how do you multiply out (x+deltax)^.5?

 

[SammyS]

So, do you need to find the derivative by finding the limit: $\displaystyle \lim_{\Delta x \to 0} \frac{F(x+\Delta x)-F(x)}{\Delta x}\,?$

If so, you'll find it handy to rational the numerator.

 

[jmanna98]

Yes as delta x approaches zero. I know there is goig to be some conjugate or LCD stuff going on but i got stuck

 

[HallsofIvy]

So far you haven't shown any work at all. What have you tried to do and where, exactly, do you have a problem?

 

[jmanna98]

The first thing I did was sub the function into the derivative formula which made a huge mess of a problem to simplify.

[(-1/sqrt2(x+deltax)) +2(x+deltax)] - [(-1/sqrt2x)+2x] all over deltax.

I am a little rusty on working with radicals and tried a few things that ended up in a mess but I am thinking that I should simplify the numerator of this first by finding the LCD of the rational expressions in the numberator of the whole problem. LCD:(sqrt2(x+deltax))(sqrt2x)? Then multiply by the conjugate? I sort of feel on the right track but at the same time I feel that my LCD is incorrect for some reason.

 

[SammyS]

The first thing I did was sub the function into the derivative formula which made a huge mess of a problem to simplify.

[(-1/sqrt2(x+deltax)) +2(x+deltax)] - [(-1/sqrt2x)+2x] all over deltax.

I am a little rusty on working with radicals and tried a few things that ended up in a mess but I am thinking that I should simplify the numerator of this first by finding the LCD of the rational expressions in the numberator of the whole problem. LCD:(sqrt2(x+deltax))(sqrt2x)? Then multiply by the conjugate? I sort of feel on the right track but at the same time I feel that my LCD is incorrect for some reason.

Let's look at your difference quotient in LaTeX.
[(-1/sqrt2(x+deltax)) +2(x+deltax)] - [(-1/sqrt2x)+2x] all over deltax
$$\frac{\displaystyle \frac{-1}{\sqrt{2(x+\Delta x)}}-\left(\frac{-1}{\sqrt{2(x)}}+2x\right)}{\Delta x}\quad\to\quad \frac{\displaystyle \frac{-1}{\sqrt{2(x+\Delta x)}}-\frac{-1}{\sqrt{2(x)}}}{\Delta x}+\frac{2(x+\Delta x) -2x}{\Delta x}$$

 

[SteamKing]

Instead of trying to find the limit of all of the parts of F(x) at one time, break F(x) into two pieces: 2x and the radical. Since the derivative of a sum is the sum of the derivatives of the components, you can calculate the two limits and add them together.