# Help finding the derivative of rational/radical function

by jmanna98
 P: 6 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
 Emeritus Sci Advisor HW Helper Thanks PF Gold P: 6,533 You can rewrite F(x) = 2x - (2x)^(-1/2) Try taking the derivative of this expression.
 P: 6 ya but then how do you multiply out (x+deltax)^.5?
 Emeritus Sci Advisor HW Helper PF Gold P: 7,808 Help finding the derivative of rational/radical function 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.
 P: 6 Yes as delta x approaches zero. I know there is goig to be some conjugate or LCD stuff going on but i got stuck
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,552 So far you haven't shown any work at all. What have you tried to do and where, exactly, do you have a problem?
 P: 6 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 sorta feel on the right track but at the same time I feel that my LCD is incorrect for some reason.
Emeritus
$$\frac{\displaystyle \frac{-1}{\sqrt{2(x+\Delta x)}}-\left(\frac{-1}{\sqrt{2(x)}}+2x\right)}{\Delta x}\quad\to\quad \frac{\displaystyle \frac{-1}{\sqrt{2(x+\Delta x)}}-\frac{-1}{\sqrt{2(x)}}}{\Delta x}+\frac{2(x+\Delta x) -2x}{\Delta x}$$