Question

how do you find the derivative of f(t)=t-sqrt(t) by the definition of a derivative.

Answer 1

\begin{eqnarray*} \frac{f(t + h) - f(t)}{h} &=& \frac{t + h - \sqrt{t + h} - (t - \sqrt{t})}{h}\\ &=& \frac{h + \sqrt{t} - \sqrt{t + h}}{h} \\ &=& 1 - \frac{\sqrt{t + h} - \sqrt{t}}{h} \\ &=& 1 - \frac{t + h - t}{h(\sqrt{t + h} + \sqrt{t})} \\&=& 1 - \frac{1}{\sqrt{t + h} + \sqrt{t}} \\&\xrightarrow{h\ \to\ 0}& 1 - \frac{1}{\sqrt{t + 0} + \sqrt{t}} \\ &=& 1- \frac{1}{2\sqrt{t}}.\end{eqnarray*}

Answer 2

wow! thanks I feel terrible. I made a stupid mistake and got caught up