Question

section 2.4: power and sum rules for d
score: 80/180 answered: 8/18
question 9
find the derivative of: $10sqrt{x}+\frac{1}{x^{9}}$.
type your answer without fractional or negative exponents.

Explanation:

Step1: Rewrite the terms

Rewrite $10\sqrt{x}+\frac{1}{x^{9}}$ as $10x^{\frac{1}{2}}+x^{- 9}$.

Step2: Apply power - rule for derivatives

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$.
For $y_1 = 10x^{\frac{1}{2}}$, $y_1^\prime=10\times\frac{1}{2}x^{\frac{1}{2}-1}=5x^{-\frac{1}{2}}$.
For $y_2 = x^{-9}$, $y_2^\prime=-9x^{-9 - 1}=-9x^{-10}$.

Step3: Combine the derivatives

The derivative of $y = 10x^{\frac{1}{2}}+x^{-9}$ is $y^\prime=y_1^\prime + y_2^\prime=5x^{-\frac{1}{2}}-9x^{-10}$.

Step4: Rewrite without fractional and negative exponents

$y^\prime=\frac{5}{\sqrt{x}}-\frac{9}{x^{10}}$.

Answer:

$\frac{5}{\sqrt{x}}-\frac{9}{x^{10}}$