Q. 57 - Calculus

Question

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.      

$f\left(x\right)=\sqrt{\frac{1}{x^3}}+{\left(\frac{1}{\sqrt{x}}\right)}^3$

Step-by-Step Solution

Answer

The derivative of the function is $-3x^{\frac{-5}{2}}$.

Step 1. Given Information

The given function is $f\left(x\right)=\sqrt{\frac{1}{x^3}}+{\left(\frac{1}{\sqrt{x}}\right)}^3$.

Step 2. Simplify

Simplify the given function.

$\begin{array}{rcl}f\left(x\right)& =& \frac{1}{x^{{\displaystyle \frac{3}{2}}}}+\frac{1}{x^{{\displaystyle \frac{3}{2}}}}\\ & =& \frac{2}{x^{{\displaystyle \frac{3}{2}}}}\\ & =& 2x^{-\frac{3}{2}}\end{array}$

Step 3. Find the derivative

• Apply the constant multiple of the derivative, $\left(kf\right)\text{'}\left(x\right)=kf\text{'}\left(x\right)$.

$f\text{'}\left(x\right)=2\frac{d}{dx}\left(x^{\frac{-3}{2}}\right)$

• Apply the power rule of derivative.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& 2\left(-\frac{3}{2}{x}^{-\frac{3}{2}-1}\right)\\ & =& -3x^{\frac{-5}{2}}\end{array}$