What is the derivative of this function?

\(g(t) = (t^3+3t^2+t)/(t^3)\)

I'm not getting how the extended power rule works

Thanks for any help

CurlyFry
Mar 26, 2018

#2**+2 **

**What is the derivative of this function?**

**\(g(t) = (t^3+3t^2+t)/(t^3)\)**

g(t) = (t^3+3t^2+t)/(t^3)

**I'm not getting how the extended power rule works**

**Formula power rule:**

\(\begin{array}{|rcll|} \hline f(x) &=& x^n,\quad n \ne 0 \\ f'(x) &=& n\cdot x^{n-1} \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline g(t) &=& \dfrac{t^3+3t^2+t} {t^3} \\\\ &=& \dfrac{t^3} {t^3} +\dfrac{3t^2} {t^3}+\dfrac{t} {t^3} \\\\ &=& 1 + 3\cdot t^{2-3} + t^{1-3} \\\\ &=& 1 + 3\cdot t^{-1} + t^{-2} \quad & | \quad \text{power rule} \\\\ g'(x) &=& 0 + 3\cdot \left((-1)\cdot t^{-1-1} \right)+ (-2) \cdot t^{-2-1} \\\\ &=& -3\cdot t^{-2} - 2 \cdot t^{-3} \\\\ &=& - \dfrac{3}{t^2} - \dfrac{2}{t^3} \\\\ \hline \end{array} \)

heureka
Mar 26, 2018