Question

find the derivative of the following function by first simplifying the expression. f(w) = \frac{5w^{5}-3w}{7w} f(w) =

Explanation:

Step1: Simplify the function

First, divide each term in the numerator by \(7w\):
\[

$$\begin{align*} f(w)&=\frac{5w^{5}-3w}{7w}\\ &=\frac{5w^{5}}{7w}-\frac{3w}{7w}\\ &=\frac{5}{7}w^{4}-\frac{3}{7} \end{align*}$$

\]

Step2: Apply the power - rule for differentiation

The power - rule states that if \(y = ax^{n}\), then \(y^\prime=anx^{n - 1}\), and the derivative of a constant \(C\) is \(0\).
For \(y=\frac{5}{7}w^{4}-\frac{3}{7}\), the derivative of \(\frac{5}{7}w^{4}\) is \(\frac{5}{7}\times4w^{4 - 1}=\frac{20}{7}w^{3}\), and the derivative of \(-\frac{3}{7}\) (a constant) is \(0\).

Answer:

\(\frac{20}{7}w^{3}\)