Question

In Exercises \(1-6,\) find \(d y / d x\). $$y=\frac{x^{3}}{3}+\frac{x^{2}}{2}+x$$

Short answer

The derivative of the function \(y = \frac{x^{3}}{3} + \frac{x^{2}}{2} + x\) is \(d y / d x = x^2 + x + 1\).

Step-by-step solution

Identify Various Parts of the Function

The function provided is actually a polynomial, which means it consists of several terms - in this case, three: \(\frac{x^{3}}{3}\), \(\frac{x^{2}}{2}\), and \(x\). Each of these will have to be derived independently according to the power rule.

Apply the Power Rule to the First Term

For the first term \(\frac{x^{3}}{3}\), based on the power rule, the derivative is \(3 \cdot \frac{1}{3} \cdot x^{3-1}\), which simplifies to \(x^2\).

Apply the Power Rule to the Second Term

Similarly, for the second term, \(\frac{x^{2}}{2}\), the derivative based on the power rule is \(2 \cdot \frac{1}{2} \cdot x^{2-1}\), which simplifies to \(x\).

Apply The Power Rule to The Third Term

Lastly, for the final term, \(x\), the derivative is \(1 \cdot x^{1-1}\), which simplifies to \(1\).

Combine the Derived Parts

Combining the derived parts of the function, we get the derivatives of the first, second, and third terms to be \(x^2\), \(x\), and \(1\), respectively. Thus, the derivative of the whole function, \(d y / d x\), is \(x^2 + x + 1\).