Use the Power Rule to compute the following derivatives. You may need to simplify the expression before differentiating. (a) \( \frac{d}{d x} \frac{1}{x^{2}} \) (b) \( \frac{d}{d x} x^{x} \) (c) \( \frac{d}{d x} \frac{x^{9}}{x^{3}} \) (a) \( \frac{d}{d x} \frac{1}{x^{2}}=\square \)

To compute \( \frac{d}{d x} \frac{1}{x^{2}} \), we can first rewrite the expression: \( \frac{1}{x^{2}} = x^{-2} \). Now, applying the Power Rule: \[ \frac{d}{d x} x^{-2} = -2x^{-3} = -\frac{2}{x^{3}}. \] So, \( \frac{d}{d x} \frac{1}{x^{2}} = -\frac{2}{x^{3}} \). --- (b) For \( \frac{d}{d x} x^{x} \), we will need to use logarithmic differentiation. Let \( y = x^{x} \). Taking the natural logarithm of both sides gives: \[ \ln(y) = x \ln(x). \] Now, differentiate both sides with respect to \( x \): \[ \frac{1}{y} \frac{dy}{dx} = \ln(x) + 1. \] Thus, \[ \frac{dy}{dx} = y(\ln(x) + 1) = x^{x}(\ln(x) + 1). \] So \( \frac{d}{d x} x^{x} = x^{x}(\ln(x) + 1) \). --- (c) To compute \( \frac{d}{d x} \frac{x^{9}}{x^{3}} \), simplify the expression first: \[ \frac{x^{9}}{x^{3}} = x^{6}. \] Now apply the Power Rule: \[ \frac{d}{d x} x^{6} = 6x^{5}. \] So \( \frac{d}{d x} \frac{x^{9}}{x^{3}} = 6x^{5} \).