find the derivative of the following function by first simplifying the expression. f(w)=(2w^9 - 7w^5)/(9w^5)…

Explanation:

Step1: Simplify the function

[ \begin{align*} f(w)&=\frac{2w^{9}-7w^{5}}{9w^{5}}\ &=\frac{2w^{9}}{9w^{5}}-\frac{7w^{5}}{9w^{5}}\ &=\frac{2}{9}w^{9 - 5}-\frac{7}{9}\ &=\frac{2}{9}w^{4}-\frac{7}{9} \end{align*} ]

Step2: Differentiate term - by - term

The derivative of a constant $C$ is $0$, and the power rule for differentiation is $\frac{d}{dw}(aw^{n})=naw^{n - 1}$. For the first term $\frac{2}{9}w^{4}$, using the power rule: $\frac{d}{dw}(\frac{2}{9}w^{4}) = 4\times\frac{2}{9}w^{4 - 1}=\frac{8}{9}w^{3}$. For the second term $-\frac{7}{9}$ (a constant), $\frac{d}{dw}(-\frac{7}{9}) = 0$.

Answer:

$\frac{8}{9}w^{3}$