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# Extended power rule?

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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
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+9515
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What is the derivative of this function?

Omi67  Mar 26, 2018
#2
+19813
+3

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