https://file.aiquickdraw.com/mathbot/file/1730003671337ojuq15a9.jpg

To find \(\frac{dy}{dx}\) by implicit differentiation, let's start with the given equation:

\[
\frac{e^{x}}{y} = 8x - y
\]

### Step 1: Differentiate both sides with respect to \(x\)

Since \(y\) is a function of \(x\), we will need to apply the chain rule when differentiating any term involving \(y\).

Differentiate the left side:
\[
\frac{d}{dx} \left( \frac{e^{x}}{y} \right)
\]

Use the quotient rule \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\), where \(u = e^x\) and \(v = y\):
\[
u' = e^{x}, \quad v' = \frac{dy}{dx}
\]
\[
\frac{d}{dx} \left( \frac{e^{x}}{y} \right) = \frac{e^{x} \cdot y - e^{x} \cdot \frac{dy}{dx}}{y^2}
\]

Differentiate the right side:
\[
\frac{d}{dx}(8x - y) = 8 - \frac{dy}{dx}
\]

### Step 2: Equate the derivatives

\[
\frac{e^{x}y - e^{x}\frac{dy}{dx}}{y^2} = 8 - \frac{dy}{dx}
\]

### Step 3: Solve for \(\frac{dy}{dx}\)

Multiply both sides by \(y^2\) to eliminate the denominator:
\[
e^{x}y - e^{x}\frac{dy}{dx} = (8 - \frac{dy}{dx})y^2
\]

Expand and rearrange to isolate \(\frac{dy}{dx}\):
\[
e^{x}y - 8y^2 = e^{x}\frac{dy}{dx} - y^2\frac{dy}{dx}
\]

Factor out \(\frac{dy}{dx}\) on the right side:
\[
e^{x}y - 8y^2 = \frac{dy}{dx}(e^{x} - y^2)
\]

Divide both sides by \(e^{x} - y^2\):
\[
\frac{dy}{dx} = \frac{e^{x}y - 8y^2}{e^{x} - y^2}
\]

Thus, the derivative \(\frac{dy}{dx}\) is:
\[
\boxed{\frac{dy}{dx} = \frac{e^{x}y - 8y^2}{e^{x} - y^2}}
\]

Would you like any additional details or explanations?

### Relative Questions:
1. What are the conditions under which implicit differentiation is necessary?
2. Can you explain the quotient rule used for differentiating \(\frac{e^x}{y}\)?
3. How does implicit differentiation differ from explicit differentiation?
4. Could you solve for \(y\) explicitly in this equation and then differentiate?
5. What other scenarios require the chain rule in differentiation?

### Tip:
When differentiating implicitly, always remember to treat \(y\) as a function of \(x\), applying the chain rule wherever \(y\) appears.

Find \( \frac{dy}{dx} \) by implicit differentiation given that \( \frac{e^x}{y} = 8x - y \).

Learn how to find \( \frac{dy}{dx} \) using implicit differentiation for the equation \( \frac{e^x}{y} = 8x - y \). This solution involves applying the quotient rule and chain rule, making it suitable for advanced calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation