Question

section 2.4: power and sum rules for
score: 60/180 answered: 6/18
question 7
find the derivative of: -3x^{1/6}+2x^{-10}
question help: video

Explanation:

Step1: Apply power - rule to first term

The power - rule for differentiation is $\frac{d}{dx}(ax^n)=nax^{n - 1}$. For the term $-3x^{\frac{1}{6}}$, where $a=-3$ and $n = \frac{1}{6}$, we have $\frac{d}{dx}(-3x^{\frac{1}{6}})=\frac{1}{6}\times(-3)x^{\frac{1}{6}-1}=-\frac{1}{2}x^{-\frac{5}{6}}$.

Step2: Apply power - rule to second term

For the term $2x^{-10}$, where $a = 2$ and $n=-10$, we have $\frac{d}{dx}(2x^{-10})=-10\times2x^{-10 - 1}=-20x^{-11}$.

Step3: Use sum - rule of differentiation

The sum - rule states that $\frac{d}{dx}(u + v)=\frac{du}{dx}+\frac{dv}{dx}$. Here $u=-3x^{\frac{1}{6}}$ and $v = 2x^{-10}$. So the derivative of $-3x^{\frac{1}{6}}+2x^{-10}$ is $-\frac{1}{2}x^{-\frac{5}{6}}-20x^{-11}$.

Answer:

$-\frac{1}{2}x^{-\frac{5}{6}}-20x^{-11}$