**Using the Power Rule to Compute Derivatives**The Power Rule for differentiation states that if you have a function $ f(x) = x^n $, then its derivative $ f'(x) $ is given by:\[ f'(x) = nx^{n-1} \]**Example Problem:**Use the Power Rule to compute the derivative of the function and evaluate it at a given point.\[ \frac{d}{dt} t^{-2} \Bigg|_{t=9} = \]**Steps for Solution:**1. **Apply the Power Rule:** For the function $ t^{-2} $, we can apply the Power Rule where $ n = -2 $: \[ \frac{d}{dt} t^{-2} = -2t^{-3} \]2. **Evaluate at $ t = 9 $:** We need to substitute $ t = 9 $ into the derivative $ -2t^{-3} $: \[ -2(9)^{-3} = -2 \cdot \frac{1}{9^3} = -2 \cdot \frac{1}{729} = -\frac{2}{729} \]Therefore:\[ \frac{d}{dt} t^{-2} \Bigg|_{t=9} = -\frac{2}{729} \]

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Description: **Using the Power Rule to Compute Derivatives**The Power Rule for differentiation states that if you have a function $ f(x) = x^n $, then its derivative $ f'(x) $ is given by:\[ f'(x) = nx^{n-1} \]**Example Problem:**Use the Power Rule to comput

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Calculus: Early Transcendentals

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ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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**Using the Power Rule to Compute Derivatives**

The Power Rule for differentiation states that if you have a function $ f(x) = x^n $, then its derivative $ f'(x) $ is given by:

\[ f'(x) = nx^{n-1} \]

**Example Problem:**

Use the Power Rule to compute the derivative of the function and evaluate it at a given point.

\[ \frac{d}{dt} t^{-2} \Bigg|_{t=9} = \]

**Steps for Solution:**

1. **Apply the Power Rule:**

For the function $ t^{-2} $, we can apply the Power Rule where $ n = -2 $:

\[ \frac{d}{dt} t^{-2} = -2t^{-3} \]

2. **Evaluate at $ t = 9 $:**

We need to substitute $ t = 9 $ into the derivative $ -2t^{-3} $:

\[ -2(9)^{-3} = -2 \cdot \frac{1}{9^3} = -2 \cdot \frac{1}{729} = -\frac{2}{729} \]

Therefore:

\[ \frac{d}{dt} t^{-2} \Bigg|_{t=9} = -\frac{2}{729} \]

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