fractions & derivatives

**becky** · May 7th 2006, 11:35 AM

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.

**Jameson** · May 7th 2006, 12:20 PM

This belongs in the Calculus section, not the elementary math section.

Ok. So using the definition of a derivative $f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{f(t+\Del ta{t})-f(t)}{\Delta{t}}$ .
Thus for finding the derivative of $\frac{2}{t}$ we find that:

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

Multiply the numerator and denominator by $(t+\Delta{t})$

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

Split up the numerator to cancel some things:

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

And from here it's just algebra.

$f'(t)=\lim_{\Delta{t}\rightarrow{0}}\frac{-2\Delta{t}}{t}*\frac{1}{\Delta{t}(t+\Delta{t})}$

Distribute and apply the limit:

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

And once you get your answer, just plug in for $t=\frac{1}{2}$

EDIT: For some reason, the Latex on the definition of the limit isn't working, but I think you're familiar with it.

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