6. -/1 points details my notes differentiate. f(y)=(1/y² - 3/y⁴)(y + 9y³) need help? read it watch it master…

Explanation:

Step1: Expand the function

First, expand $F(y)=\left(\frac{1}{y^{2}}-\frac{3}{y^{4}}\right)(y + 9y^{3})$ using the distributive - property. $F(y)=\frac{1}{y^{2}}\times y+\frac{1}{y^{2}}\times9y^{3}-\frac{3}{y^{4}}\times y-\frac{3}{y^{4}}\times9y^{3}$ $F(y)=y^{-1}+9y - 3y^{-3}-27y^{-1}$ $F(y)=9y-26y^{-1}-3y^{-3}$

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. For the term $9y$, its derivative is $9\times1\times y^{1 - 1}=9$. For the term $-26y^{-1}$, its derivative is $-26\times(-1)\times y^{-1 - 1}=26y^{-2}$. For the term $-3y^{-3}$, its derivative is $-3\times(-3)\times y^{-3 - 1}=9y^{-4}$.

Step3: Write the derivative of the function

$F^\prime(y)=9 + \frac{26}{y^{2}}+\frac{9}{y^{4}}$

Answer:

$9+\frac{26}{y^{2}}+\frac{9}{y^{4}}$