Q. 54 - 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.     

$f\left(x\right)={\left(3x\sqrt{x}\right)}^{-2}$

Step-by-Step Solution

Answer

The derivative of the given function is $\frac{-1}{3x^4}$.

Step 1. Given Information

The given expression is $f\left(x\right)={\left(3x\sqrt{x}\right)}^{-2}$.

Step 2. Simplify the function

Simplify the function.

$\begin{array}{rcl}f\left(x\right)& =& {\left(3x^{\frac{3}{2}}\right)}^{-2}\\ & =& 3^{-2}{x}^{-3}\\ & =& \frac{x^{-3}}{9}\end{array}$

Step 2. Find the derivative

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

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}\left(\frac{x^{-3}}{9}\right)\\ & =& \frac{1}{9}\frac{d}{dx}\left(x^{-3}\right)\end{array}$

• Apply the power rule of derivative, $\left(x^n\right)\text{'}=n{x}^{n-1}$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{1}{9}(-3x^{-4})\\ & =& \frac{-x^{-4}}{3}\\ & =& \frac{-1}{3x^4}\end{array}$