find the derivative of the function.\ny = \\frac{x^{7}+14}{x^{3}}\nsimplify y by dividing each term in the…

find the derivative of the function.\ny = \\frac{x^{7}+14}{x^{3}}\nsimplify y by dividing each term in the numerator by x^{3}.\ny = \nfind the derivative of the function.\n\\frac{dy}{dx} =

find the derivative of the function.\ny = \\frac{x^{7}+14}{x^{3}}\nsimplify y by dividing each term in the numerator by x^{3}.\ny = \nfind the derivative of the function.\n\\frac{dy}{dx} =

Answer

Explanation:

Step1: Simplify the function

Divide each term in the numerator by $x^{3}$. Using the rule $\frac{x^{m}}{x^{n}}=x^{m - n}$, we have $\frac{x^{7}}{x^{3}}=x^{7-3}=x^{4}$ and $\frac{14}{x^{3}} = 14x^{-3}$. So $y=x^{4}+14x^{-3}$.

Step2: Find the derivative

Use the power - rule for differentiation $\frac{d}{dx}(x^{n})=nx^{n - 1}$. For the first term $x^{4}$, its derivative is $4x^{4-1}=4x^{3}$. For the second term $14x^{-3}$, its derivative is $14\times(-3)x^{-3 - 1}=-42x^{-4}$. So $\frac{dy}{dx}=4x^{3}-42x^{-4}$.

Answer:

$y=x^{4}+14x^{-3}$; $\frac{dy}{dx}=4x^{3}-42x^{-4}$