Derivative from First Principles for 3x-\frac{5}{x}

[Checkfate]

Hello. I have been trying to find this derivative from first principles for at least a couple hours, but just can't make any progress with it.

Find the derivative of $$3x-\frac{5}{x}$$

Well I start by saying that the derivative is the limit as h approaches 0 of

$$\frac{f(x+h)-f(x)}{h}$$ where h=deltax

Then I go on to say that this is equal to the limit as h approaches 0 of
$$\frac{3h-\frac{5}{x+h}+\frac{5}{x}}{h}$$

I then simplify by taking h out of the numerator by factoring and then cancel h on the numerator and denominator. This the derivative equals the limit as h approches 0 of
$$3-\frac{5}{(x+h)h}+\frac{5}{xh}$$

As you can see, my derivative is now a bloody mess and I see no way of getting h out of the denominator. Please help! By the way I need to do this from first principles (dy/dx=(f(x+h)-f(x))/(h) ). Thankyou!

 

[neutrino]

Then I go on to say that this is equal to the limit as h approaches 0 of
$$\frac{3h-\frac{5}{x+h}+\frac{5}{h}}{h}$$

The last term in the numrator is $$\frac{5}{x}$$.

 

[dmoravec]

try combining the fractions into:
$$\frac{-5x+5(x+h)}{(x+h)hx}$$
EDIT: can't seem to get that latex to work... anyone see where I went wrong with it?