Question

section 2.4: power and sum rules for deriv score: 90/180 answered: 9/18 question 10 find the derivative of: $\frac{42x^{11}+30x^{6}+6}{6x^{2}}$. type your answer without negative exponents. simplify completely

Explanation:

Step1: Simplify the function

First, divide each term in the numerator by the denominator: $\frac{42x^{11}+30x^{6}+6}{6x^{2}}=\frac{42x^{11}}{6x^{2}}+\frac{30x^{6}}{6x^{2}}+\frac{6}{6x^{2}} = 7x^{9}+5x^{4}+x^{- 2}$.

Step2: Apply power - rule for derivatives

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$.
For $y = 7x^{9}$, $y^\prime=7\times9x^{9 - 1}=63x^{8}$.
For $y = 5x^{4}$, $y^\prime=5\times4x^{4 - 1}=20x^{3}$.
For $y = x^{-2}$, $y^\prime=-2x^{-2 - 1}=-2x^{-3}$. But we need to write the answer without negative exponents, so $y^\prime = 63x^{8}+20x^{3}-\frac{2}{x^{3}}$.

Answer:

$63x^{8}+20x^{3}-\frac{2}{x^{3}}$