find the derivative of the function. y = 17t / (t ^ (2/3)) y = □

Explanation:

Step1: Rewrite the function

Rewrite $y = \frac{17t}{\sqrt[3]{t^{2}}}$ as $y=17t\cdot t^{-\frac{2}{3}} = 17t^{1-\frac{2}{3}}=17t^{\frac{1}{3}}$ using the rule $\frac{1}{a^{n}}=a^{-n}$ and $a^{m}\cdot a^{n}=a^{m + n}$.

Step2: Apply the power - rule

The power - rule for differentiation is $\frac{d}{dt}(at^{n})=ant^{n - 1}$, where $a = 17$ and $n=\frac{1}{3}$. So $y'=17\times\frac{1}{3}t^{\frac{1}{3}-1}$.

Step3: Simplify the exponent

Calculate $\frac{1}{3}-1=\frac{1 - 3}{3}=-\frac{2}{3}$. Then $y'=\frac{17}{3}t^{-\frac{2}{3}}=\frac{17}{3t^{\frac{2}{3}}}$.

Answer:

$\frac{17}{3t^{\frac{2}{3}}}$