Questions: Let y = ∛x * (1/x - 2). Use the product rule to compute and simplify y′.

Transcript text: this exam. 1. (2 pts) Let $y=\sqrt[3]{x} \cdot\left(\frac{1}{x}-2\right)$. Use the product rule to compute and simplify $y^{\prime}$.

Solution

Solution Steps

To find the derivative $ y' $ of the function $ y = \sqrt[3]{x} \cdot \left(\frac{1}{x} - 2\right) $ using the product rule, we need to:

Identify the two functions being multiplied: $ u = \sqrt[3]{x} $ and $ v = \frac{1}{x} - 2 $.
Compute the derivatives $ u' $ and $ v' $.
Apply the product rule: $ y' = u'v + uv' $.

Step 1: Define the Functions

Let $ y = \sqrt[3]{x} \cdot \left(\frac{1}{x} - 2\right) $. We define:

$ u = \sqrt[3]{x} = x^{1/3} $
$ v = \frac{1}{x} - 2 $

Step 2: Compute the Derivatives

We calculate the derivatives of $ u $ and $ v $:

$ u' = \frac{1}{3} x^{-2/3} $
$ v' = -\frac{1}{x^2} $

Step 3: Apply the Product Rule

Using the product rule $ y' = u'v + uv' $: \[ y' = \left(\frac{1}{3} x^{-2/3}\right) \left(\frac{1}{x} - 2\right) + \left(\sqrt[3]{x}\right) \left(-\frac{1}{x^2}\right) \]

Step 4: Simplify the Expression

Substituting the derivatives into the product rule gives: \[ y' = -\frac{1}{x^{5/3}} + \frac{1}{3} \left(\frac{1}{x} - 2\right) x^{-2/3} \] This simplifies to: \[ y' = -\frac{1}{x^{5/3}} + \frac{1}{3} \left(\frac{1}{x^{5/3}} - \frac{2}{x^{2/3}}\right) \] Combining the terms results in: \[ y' = -\frac{1}{x^{5/3}} + \frac{1}{3x^{5/3}} - \frac{2}{3x^{2/3}} = -\frac{2}{3x^{2/3}} - \frac{2}{3x^{5/3}} \]