Q. 62 - 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.  

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

Step-by-Step Solution

Answer

The derivative of the function is $\begin{array}{rcl}& & \frac{{\displaystyle -3}}{(x+1)}\end{array}$.

Step 1. Given Information

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

Step 2. Find the derivative

Apply the quotient rule of derivative, $\left(\frac{f}{g}\right)\text{'}\left(x\right)=\frac{f\text{'}\left(x\right)g\left(x\right)-f\left(x\right)g\text{'}\left(x\right)}{(g{\left(x\right))}^2}$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{{\displaystyle \frac{d}{dx}}\left(1\right)\times {(x+1)}^3-\left(1\right){\displaystyle \frac{d}{dx}}{(x+1)}^3}{{(x+1)}^3}\\ & =& \frac{{\displaystyle \left(0\right)\times {(x+1)}^3-3{(x+1)}^2}}{{(x+1)}^3}\\ & =& \frac{{\displaystyle -3}}{(x+1)}\end{array}$