Use the Power Rule to compute the following derivatives. You ma (a) \( \frac{d}{d x} \frac{1}{x^{2}} \) (b) \( \frac{d}{d x} x^{\pi} \) (c) \( \frac{d}{d x} \frac{x^{9}}{x^{3}} \) (u) \( d x x^{2} \) (b) \( \frac{d}{d x} x^{\pi}=\pi x^{\pi-1} \) (c) \( \frac{d}{d x} x^{9} x^{3}=\square \)

**(a) Derivative of \(\frac{1}{x^2}\):** Write the function as \[ \frac{1}{x^2} = x^{-2}. \] Using the Power Rule, where if \(f(x)=x^n\) then \(f'(x)=n x^{n-1}\), we have \[ \frac{d}{dx}\,x^{-2}= -2\, x^{-2-1} = -2\, x^{-3}. \] This can also be written as \[ -\frac{2}{x^3}. \] --- **(b) Derivative of \(x^{\pi}\):** For the function \[ f(x)=x^{\pi}, \] the Power Rule gives \[ f'(x)=\pi\, x^{\pi-1}. \] --- **(c) Derivative of \(\frac{x^{9}}{x^{3}}\):** First, simplify the function using the quotient of powers: \[ \frac{x^9}{x^3}=x^{9-3}=x^6. \] Then, apply the Power Rule: \[ \frac{d}{dx}\,x^6=6\,x^{6-1}=6\,x^5. \] --- **Additional Derivatives (if needed):** 1. **Derivative of \(x^2\):** \[ \frac{d}{dx}\,x^2=2\,x. \] 2. **Derivative of \(x^9 \cdot x^3\):** Recognize that \[ x^9 \cdot x^3 = x^{9+3} = x^{12}. \] Then, \[ \frac{d}{dx}\,x^{12}=12\, x^{12-1}=12\,x^{11}. \]