What is the derivative of F(x) = 2x - 3/x by the Definition of a Derivative

I've gone through it several times and haven't been able to come out with the correct derivative, or its equivalent. The actual derivative by derivative rules is 2 - 3/x^2

I always hated these; if I could design a calculus curriculum, I'd hold off on the derivative definition until after I taught infinite series and convergence. They do kind of meet you halfway and teach you newton's method around then, but I digress.

So the derivative definition is

lim h->0 [f(x+h) + f(x)]/h .

Remember this, and it will make your life easier: When you see

lim h->0

That's a fancy way of saying this:

Do all the algebra first, then substitute this number in for h.

So let's set up the functions. f(x) is easy; it's just F(x). So let's do f(x+h),

2(x+h) - 3/(x+h) = 2x + 2h - 3/(x+h)

Now we can sub that whole thing back in,

[ (2x + 2h - 3/(x+h)) - (2x - 3/x) ] / h = [ 2h - 3/(x+h) + 3/x ] / h

Now, let's combine those two fractional parts. The least common multiple for x+h and x is x(x+h), so that becomes

[ 2h - 3x/x(x+h) + 3(x+h)/x(x+h) ] / h

or

[ 2h - (3x + 3h - 3x) / (x(x+h) ] / h

or

[ 2h - 3h / (x(x+h)) ] /h

Now we just divide h through,

2 - 3 / (x(x+h))

And now that we're completely done with all the algebra, we sub h back in, to get the right answer,

2 - 3/x(x+0) = 2 - 3 / x^2

The step that was probably tripping you up was simplifying the two fractions. Hope this helped.

well, you can change the 3/x into 3x^-1, then find the derivative (of the whole thing) which would be 2-(-3x^-2)
that simplifies down into the correct answer since you can move the x^-2 into the denominator