Q. 45 - 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.    

$f\left(x\right)=\frac{1-6x^3}{3}$

Step-by-Step Solution

Answer

The derivative of the function is $\begin{array}{rcl}f\text{'}\left(x\right)& =& -6x^2\end{array}$.

Step 1. Given Information

The given function is $f\left(x\right)=\frac{1-6x^3}{3}$.

Step 2. Simplify the function

Simplify the given function.

$\begin{array}{rcl}f\left(x\right)& =& \frac{1}{3}-\frac{6x^3}{3}\\ & =& \frac{1}{3}-2x^3\end{array}$

Step 3. Find the derivative

• Apply the difference rule of derivative, $(f-g)\text{'}\left(x\right)=f\text{'}\left(x\right)-g\text{'}\left(x\right)$.

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

• Apply the constant multiple rule, $\left(kf\right)\text{'}\left(x\right)=kf\text{'}\left(x\right)$ and the derivative of the constant function, $f\text{'}\left(k\right)=0$ in the derived function.

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

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

$\begin{array}{rcl}f\text{'}\left(x\right)& =& -2\left(3x^2\right)\\ & =& -6x^2\end{array}$