1. ## Difference Quotient

I'm new to calculus and haven't had pre-calc. I have a question regarding a certain problem.

I'm to compute the derivative of a given function using the difference quotient.

This is the function: $\displaystyle f(x)= -2/x$

I only got as far as plugging the function into the formula before I got lost. Can someone please help?

2. Originally Posted by gstyzzer
I'm new to calculus and haven't had pre-calc. I have a question regarding a certain problem.

I'm to compute the derivative of a given function using the difference quotient.

This is the function: $\displaystyle f(x)= -2/x$

I only got as far as plugging the function into the formula before I got lost. Can someone please help?
Let $\displaystyle f(x) = \frac{-2}{x}$, then:

$\displaystyle \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\frac{-2}{x+h} - \frac{-2}{x}}{h}$ $\displaystyle = \lim_{h \to 0} \frac{\frac{-2x}{x(x+h)} + \frac{2(x+h)}{x(x+h)}}{h}$

Can you finish it off from here? (Rest is in the spoiler.)

Spoiler:
$\displaystyle \lim_{h \to 0} \frac{-2x +2x +2h}{hx(x+h)} = \lim_{h \to 0} \frac{2}{x(x+h)} = \lim_{h \to 0} \frac{2}{x^2+xh} = \frac{2}{x^2}$

f'(x)=g(x)h'(x)-h(x)g'(x)
[g(x)]^2

let -2=h(x) and x=g(x)
f(x)=h(x)/g(x)

f'(x)={xd(-2)-(-2)d(x)}/(x)^2
dx d(x)
=0+2
x^2
=2/x^2