
fractions & derivatives
I have forgotten some of the simplest things from my school days.
How would I go about getting rid of the denominator in a problem such as 2/t.
How would I solve a problem when there is a fraction in a fraction? I am doing derivatives and the actual problem is 2/t if t=1/2.
Using the formula f(x) lim h>0 f(x+h)  f(x)/h.

This belongs in the Calculus section, not the elementary math section. :D
Ok. So using the definition of a derivative $\displaystyle f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{f(t+\Del ta{t})f(t)}{\Delta{t}}$.
Thus for finding the derivative of $\displaystyle \frac{2}{t}$ we find that:
$\displaystyle f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{\frac{2} {t+\Delta{t}}\frac{2}{t}}{\Delta{t}}$
Multiply the numerator and denominator by $\displaystyle (t+\Delta{t})$
$\displaystyle f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{2\left(\frac{2t+2\Delta{t}}{t}\right)}{\Delta{t}(t+ \Delta{t})}$
Split up the numerator to cancel some things:
$\displaystyle f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{22\frac{2\Delta{t}}{t}}{\Delta{t}(t+\Delta{t})}$
And from here it's just algebra.
$\displaystyle f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{2\Delta{t}}{t}*\frac{1}{\Delta{t}(t+\Delta{t})}$
Distribute and apply the limit:
$\displaystyle f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{2}{t^2+t\Delta{t}}$
And once you get your answer, just plug in for $\displaystyle t=\frac{1}{2}$
EDIT: For some reason, the Latex on the definition of the limit isn't working, but I think you're familiar with it.