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To find \( y' \) (the derivative of \( y \) with respect to \( x \)) by implicit differentiation, let's differentiate both sides of the equation with respect to \( x \). The equation is:

\[
e^{x/y} = 3x - 4y
\]

### Step 1: Differentiate the left-hand side

The derivative of \( e^{x/y} \) requires the chain rule:

\[
\frac{d}{dx} \left( e^{x/y} \right) = e^{x/y} \cdot \frac{d}{dx} \left( \frac{x}{y} \right)
\]

Now, use the quotient rule to differentiate \( \frac{x}{y} \):

\[
\frac{d}{dx} \left( \frac{x}{y} \right) = \frac{y \cdot 1 - x \cdot y'}{y^2} = \frac{y - x y'}{y^2}
\]

Thus, the derivative of the left-hand side is:

\[
e^{x/y} \cdot \frac{y - x y'}{y^2}
\]

### Step 2: Differentiate the right-hand side

The derivative of \( 3x - 4y \) is:

\[
\frac{d}{dx}(3x - 4y) = 3 - 4y'
\]

### Step 3: Combine the derivatives

Now, we combine the results of both sides:

\[
e^{x/y} \cdot \frac{y - x y'}{y^2} = 3 - 4y'
\]

### Step 4: Solve for \( y' \)

First, multiply both sides by \( y^2 \) to simplify:

\[
e^{x/y} (y - x y') = y^2 (3 - 4y')
\]

Expand both sides:

\[
e^{x/y} y - e^{x/y} x y' = 3y^2 - 4y^2 y'
\]

Now, collect all the terms involving \( y' \) on one side:

\[
- e^{x/y} x y' + 4y^2 y' = 3y^2 - e^{x/y} y
\]

Factor out \( y' \):

\[
y'(- e^{x/y} x + 4y^2) = 3y^2 - e^{x/y} y
\]

Finally, solve for \( y' \):

\[
y' = \frac{3y^2 - e^{x/y} y}{4y^2 - e^{x/y} x}
\]

This is the expression for \( y' \) found by implicit differentiation.

Would you like further clarification or details?

### Follow-up questions:
1. Can you explain the quotient rule used in this context?
2. What is the chain rule, and how was it applied to the term \( e^{x/y} \)?
3. Why did we factor out \( y' \) in the final step?
4. How do you use implicit differentiation in other types of equations?
5. Can implicit differentiation be used for parametric equations?

### Tip:
Always remember to use the chain rule when differentiating functions with variables inside an exponent!

Find y' by implicit differentiation. e^{x/y} = 3x - 4y

This detailed solution shows how to find the derivative y' using implicit differentiation for the equation e^{x/y} = 3x - 4y. The process involves applying the chain rule and quotient rule, suitable for advanced high school calculus students.

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