Chapter 2.3, Problem 2E

To determine

To calculate: The derivative $\frac{dy}{dx}$ of the function

Answer to Problem 2E

The derivative of the function is $x^2-1$ .

Explanation of Solution

Given information:

The function is $y=\frac{x^3}{3}-x$ .

Concept used:

The quotient rule is $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v.\frac{du}{dx}-u.\frac{dv}{dx}}{v^2}$ and the formula used is $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ .

Calculation:

The function is $y=\frac{x^3}{3}-x$ .

Differentiate the function with respect to $x$ .

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}\left(\frac{x^3}{3}-x\right)\\ =\frac{d}{dx}\left(\frac{x^3}{3}\right)-\frac{d}{dx}\left(x\right)\\ =\frac{3\frac{d}{dx}\left(x^3\right)-\left(x^3\right)\frac{d}{dx}\left(3\right)}{{\left(3\right)}^2}-1\\ =\frac{3.3x^2-x^3.0}{9}-1\end{array}$

Solve further.

  $\begin{array}{c}\frac{dy}{dx}=\frac{9x^2}{9}-1\\ =x^2-1\end{array}$

Conclusion: The derivative of the function is $x^2-1$