find dy for y = \\frac{4x^{2}+7}{x}.\ndy =

Explanation:

Step1: Simplify the function

First, rewrite $y = \frac{4x^{2}+7}{x}$ as $y=4x + \frac{7}{x}=4x+7x^{- 1}$.

Step2: Differentiate term - by - term

The derivative of $4x$ with respect to $x$ is $4$ (since $\frac{d}{dx}(ax)=a$ for a constant $a$), and the derivative of $7x^{-1}$ with respect to $x$ is $7\times(-1)x^{-2}=- \frac{7}{x^{2}}$ (using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$). So, $\frac{dy}{dx}=4-\frac{7}{x^{2}}$.

Step3: Find $dy$

Since $dy=\frac{dy}{dx}dx$, then $dy=(4 - \frac{7}{x^{2}})dx$.

Answer:

$(4-\frac{7}{x^{2}})dx$