Question

In Exercises \(33-36,\) find the first four derivatives of the function. $$y=x^{-1}+x^{2}$$

Short answer

The first four derivatives of the function are: \(y^{'}=-x^{-2}+2x\), \(y^{''}=2x^{-3}+2\), \(y^{'''}=-6x^{-4}\), and \(y^{''''}=24x^{-5}\).

Step-by-step solution

Identify the Function

The function given is \(y=x^{-1}+x^{2}\).

Compute the First Derivative

Using the power rule, we find the derivative of \(x^{-1}\) to be \(-x^{-2}\), and of \(x^{2}\) to be \(2x\). Thus, the first derivative is \(y^{'}=-x^{-2}+2x\).

Compute the Second Derivative

Applying the power rule again, the derivative of \(-x^{-2}\) is \(2x^{-3}\) and of \(2x\) is \(2\). So, the second derivative is \(y^{''}=2x^{-3}+2\).

Compute the Third Derivative

Differentiating again, find the derivative of \(2x^{-3}\) to be \(-6x^{-4}\) and the derivative of \(2\) is \(0\). The third derivative is \(y^{'''}=-6x^{-4}\).

Compute the Fourth Derivative

Once more, the derivative of \(-6x^{-4}\) is \(24x^{-5}\). Therefore, the fourth derivative is \(y^{''''}=24x^{-5}\).