Express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x) dx . y = 9 - S O dy = 2. dx dy 2x x2 dx O dy = dx x2 2. dy = dx 2x x2 by

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Chapter1: Functions And Models
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how do i sole. i  am getting confused or maybe i am tired of the millions of calculs assignments i get weekly. either way please help me

### Analyzing Differential Equations on an Educational Platform

#### Problem Statement:
**Objective:** Express the relationship between a small change in \( x \) and the corresponding change in \( y \) in the form \( dy = f(x) \, dx \).

Given the equation:
\[ y = 9 \sqrt{x} - \frac{8}{x} \]

#### Choices:

1. \[
\circ \quad dy = \left( \frac{9 \sqrt{x}}{2} - \frac{8}{x^2} \right) dx
\]

2. \[
\circ \quad dy = \left( \frac{9}{2 \sqrt{x}} - \frac{8}{x^2} \right) dx
\]

3. \[
\circ \quad dy = \left( \frac{9 \sqrt{x}}{2} - \frac{8}{x^2} \right) dx
\]

4. \[
\circ \quad dy = \left( \frac{9}{2 \sqrt{x}} - \frac{8}{x^2} \right) dx
\]

### Steps to Derive \( dy \):

1. **Differentiate \( y \) with respect to \( x \):**

   Given:
   \[
   y = 9 \sqrt{x} - \frac{8}{x}
   \]

2. **Apply the Differentiation Rules:**

   - The derivative of \( 9 \sqrt{x} \):
     \[
     9 \sqrt{x} = 9x^{\frac{1}{2}}
     \]
     \[
     \frac{d}{dx} (9x^{\frac{1}{2}}) = 9 \cdot \frac{1}{2} x^{\frac{-1}{2}} = \frac{9}{2} x^{-\frac{1}{2}} = \frac{9}{2 \sqrt{x}}
     \]

   - The derivative of \( - \frac{8}{x} \):
     \[
     -\frac{8}{x} = -8x^{-1}
     \]
     \[
     \frac{d}{dx} (-8x^{-1}) = -8 \cdot (-1) x^{-2} = \frac{8}{x^2}
     \
Transcribed Image Text:### Analyzing Differential Equations on an Educational Platform #### Problem Statement: **Objective:** Express the relationship between a small change in \( x \) and the corresponding change in \( y \) in the form \( dy = f(x) \, dx \). Given the equation: \[ y = 9 \sqrt{x} - \frac{8}{x} \] #### Choices: 1. \[ \circ \quad dy = \left( \frac{9 \sqrt{x}}{2} - \frac{8}{x^2} \right) dx \] 2. \[ \circ \quad dy = \left( \frac{9}{2 \sqrt{x}} - \frac{8}{x^2} \right) dx \] 3. \[ \circ \quad dy = \left( \frac{9 \sqrt{x}}{2} - \frac{8}{x^2} \right) dx \] 4. \[ \circ \quad dy = \left( \frac{9}{2 \sqrt{x}} - \frac{8}{x^2} \right) dx \] ### Steps to Derive \( dy \): 1. **Differentiate \( y \) with respect to \( x \):** Given: \[ y = 9 \sqrt{x} - \frac{8}{x} \] 2. **Apply the Differentiation Rules:** - The derivative of \( 9 \sqrt{x} \): \[ 9 \sqrt{x} = 9x^{\frac{1}{2}} \] \[ \frac{d}{dx} (9x^{\frac{1}{2}}) = 9 \cdot \frac{1}{2} x^{\frac{-1}{2}} = \frac{9}{2} x^{-\frac{1}{2}} = \frac{9}{2 \sqrt{x}} \] - The derivative of \( - \frac{8}{x} \): \[ -\frac{8}{x} = -8x^{-1} \] \[ \frac{d}{dx} (-8x^{-1}) = -8 \cdot (-1) x^{-2} = \frac{8}{x^2} \
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