differentiate each of the following: (6 pts each)\n3. $f(x)=\frac{1}{sqrt{x}}+x^{-3}$

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{1}{\sqrt{x}}+x^{-3}$ as $f(x)=x^{-\frac{1}{2}}+x^{-3}$.

Step2: Apply power - rule of differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For the first term $y_1=x^{-\frac{1}{2}}$, its derivative $y_1^\prime=-\frac{1}{2}x^{-\frac{1}{2}-1}=-\frac{1}{2}x^{-\frac{3}{2}}$. For the second term $y_2=x^{-3}$, its derivative $y_2^\prime=-3x^{-3 - 1}=-3x^{-4}$.

Step3: Find the derivative of the function

$f^\prime(x)=y_1^\prime + y_2^\prime=-\frac{1}{2}x^{-\frac{3}{2}}-3x^{-4}$.

Answer:

$f^\prime(x)=-\frac{1}{2}x^{-\frac{3}{2}}-3x^{-4}$