6. (2 points) find the derivative of each function\n a. $f(x)=-7sqrt{x}$\n b. $g(x)=\frac{3}{5x^{3}}$\n c…

Explanation:

Step1: Rewrite $f(x)$ in power - form

$f(x)=-7x^{\frac{1}{2}}$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, where $a=-7$ and $n=\frac{1}{2}$. $f^\prime(x)=\frac{1}{2}\times(-7)x^{\frac{1}{2}-1}=-\frac{7}{2}x^{-\frac{1}{2}}=-\frac{7}{2\sqrt{x}}$

Step2: Rewrite $g(x)$ in power - form

$g(x)=\frac{3}{5}x^{-3}$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, where $a = \frac{3}{5}$ and $n=-3$. $g^\prime(x)=-3\times\frac{3}{5}x^{-3 - 1}=-\frac{9}{5}x^{-4}=-\frac{9}{5x^{4}}$

Step3: Rewrite $h(x)$ in power - form

$h(x)=\frac{5}{7}x^{-\frac{2}{3}}$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, where $a=\frac{5}{7}$ and $n =-\frac{2}{3}$. $h^\prime(x)=-\frac{2}{3}\times\frac{5}{7}x^{-\frac{2}{3}-1}=-\frac{10}{21}x^{-\frac{5}{3}}=-\frac{10}{21x^{\frac{5}{3}}}$

Answer:

a. $f^\prime(x)=-\frac{7}{2\sqrt{x}}$ b. $g^\prime(x)=-\frac{9}{5x^{4}}$ c. $h^\prime(x)=-\frac{10}{21x^{\frac{5}{3}}}$