Q. 21 - Calculus

Question

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

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

Step-by-Step Solution

Answer

The derivative of $f\left(x\right)$ is:

 $f\text{'}\left(x\right)=-\frac{3x^2}{{\left(x^3+1\right)}^2}$

Step 1. Given information:

The function is:

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

Step 2. Differentiate the function with respect to x using quotient rule:

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