# fractions & derivatives

• May 7th 2006, 11:35 AM
becky
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.
• May 7th 2006, 12:20 PM
Jameson
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:

$\displaystyle f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{2-2-\frac{2\Delta{t}}{t}}{\Delta{t}(t+\Delta{t})}$

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.