Pregunta
Use the Power Rule to compute the following derivatives. You may need to (a) \( \frac{d}{d x} \frac{1}{x^{2}} \) (b) \( \frac{d}{d x} x^{\pi} \) (c) \( \frac{d}{d x} \frac{x^{9}}{x^{3}} \) (a) \( \frac{d}{d x} \frac{1}{x^{2}}=-\frac{2}{x^{3}} \) (b) \( \frac{d}{d x} x^{\pi}=\square \)
Ask by Willis George. in the United States
Mar 09,2025
Solución de inteligencia artificial de Upstudy
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The derivative of \( x^{\pi} \) is \( \pi x^{\pi-1} \).
Solución
1. For the function \( x^{\pi} \), the power rule states that
\[
\frac{d}{dx} x^n = n x^{n-1}.
\]
2. Here, \( n = \pi \). Applying the power rule, we have
\[
\frac{d}{dx} x^{\pi} = \pi x^{\pi-1}.
\]
3. Therefore, the derivative is
\[
\frac{d}{dx} x^{\pi} = \pi x^{\pi-1}.
\]
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For part (b), using the Power Rule, the derivative of \( x^{\pi} \) is \( \pi x^{\pi-1} \). So we have: (b) \( \frac{d}{d x} x^{\pi} = \pi x^{\pi-1} \) Now for part (c), simplify \( \frac{x^{9}}{x^{3}} \) to \( x^{6} \) first. Then applying the Power Rule: (c) \( \frac{d}{d x} \frac{x^{9}}{x^{3}} = \frac{d}{d x} x^{6} = 6x^{5} \)
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