To find the derivative \( \frac{dy}{dx} \) of the function \( y = x^2 - \frac{1}{x} \), we will use the rules of differentiation. Let's go through the steps in detail. ### Step 1: Identify the function The function given is: \[ y = x^2 - \frac{1}{x} \] ### Step 2: Rewrite the function To make differentiation easier, we can rewrite \( \frac{1}{x} \) as \( x^{-1} \). Thus, the function becomes: \[ y = x^2 - x^{-1} \] ### Step 3: Differentiate the function Now, we will differentiate \( y \) with respect to \( x \). We will apply the power rule of differentiation, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \). 1. Differentiate \( x^2 \): \[ \frac{d}{dx}(x^2) = 2x \] 2. Differentiate \( -x^{-1} \): \[ \frac{d}{dx}(-x^{-1}) = -(-1) \cdot x^{-2} = \frac{1}{x^2} \] ### Step 4: Combine the derivatives Now, we combine the results from the differentiation: \[ \frac{dy}{dx} = 2x + \frac{1}{x^2} \] ### Final Answer Thus, the derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = 2x + \frac{1}{x^2} \] ### Correct Option Looking at the options provided: - A. \( 2x - \frac{1}{x^2} \) - B. \( 2x + x^2 \) - C. \( 2x - x^2 \) - D. \( 2x + \frac{1}{x^2} \) The correct option is **D**. ### Explanation of Incorrect Options - **Option A: \( 2x - \frac{1}{x^2} \)**: This option incorrectly uses a negative sign for the term \( \frac{1}{x^2} \). The derivative of \( -x^{-1} \) is positive \( \frac{1}{x^2} \), not negative. - **Option B: \( 2x + x^2 \)**: This option incorrectly adds \( x^2 \) instead of \( \frac{1}{x^2} \). The term \( -\frac{1}{x} \) differentiates to \( \frac{1}{x^2} \), not \( x^2 \). - **Option C: \( 2x - x^2 \)**: This option incorrectly subtracts \( x^2 \) instead of adding \( \frac{1}{x^2} \). The derivative of \( -\frac{1}{x} \) is not related to \( -x^2 \). ### Common Pitfalls - **Misapplying the power rule**: Remember that when differentiating negative powers, the negative sign must be handled carefully. - **Forgetting to combine terms correctly**: Always ensure that you combine the results of differentiation accurately. ### Revision Summary - The derivative of \( y = x^2 - \frac{1}{x} \) is found using the power rule. - The correct derivative is \( \frac{dy}{dx} = 2x + \frac{1}{x^2} \). - The correct answer is option D. - Be cautious with signs and combining terms when differentiating.