18. [-/1 Points] DETAILS MY NOTE Find $$ \frac{d y}{d x} $$ by implicit differentiation. $$ e^{x / y}=5 x-y $$ $$ \frac{d y}{d x}=\square $$

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To find $$ \frac{d y}{d x} $$ by implicit differentiation for the equation $$ e^{x / y} = 5x - y $$, we will follow these steps:

1. Differentiate both sides of the equation with respect to $$ x $$.
2. Apply the chain rule and product rule where necessary.
3. Solve for $$ \frac{d y}{d x} $$.

Let's start by differentiating both sides of the equation:

Given:
$$
e^{x/y} = 5x - y
$$

### Step 1: Differentiate both sides

Differentiating the left side using the chain rule:
$$
\frac{d}{dx}\left(e^{x/y}\right) = e^{x/y} \cdot \frac{d}{dx}\left(\frac{x}{y}\right)
$$
Using the quotient rule for $$ \frac{x}{y} $$:
$$
\frac{d}{dx}\left(\frac{x}{y}\right) = \frac{y \cdot \frac{d}{dx}(x) - x \cdot \frac{d}{dx}(y)}{y^2} = \frac{y - x \frac{d y}{d x}}{y^2}
$$
Thus, the derivative of the left side becomes:
$$
e^{x/y} \cdot \frac{y - x \frac{d y}{d x}}{y^2}
$$

Now, differentiating the right side:
$$
\frac{d}{dx}(5x - y) = 5 - \frac{d y}{d x}
$$

### Step 2: Set the derivatives equal

Now we set the derivatives equal to each other:
$$
e^{x/y} \cdot \frac{y - x \frac{d y}{d x}}{y^2} = 5 - \frac{d y}{d x}
$$

### Step 3: Solve for $$ \frac{d y}{d x} $$

Now we will rearrange this equation to isolate $$ \frac{d y}{d x} $$.

First, multiply both sides by $$ y^2 $$:
$$
e^{x/y} (y - x \frac{d y}{d x}) = y^2(5 - \frac{d y}{d x})
$$

Expanding both sides:
$$
e^{x/y} y - e^{x/y} x \frac{d y}{d x} = 5y^2 - y^2 \frac{d y}{d x}
$$

Now, collect all terms involving $$ \frac{d y}{d x} $$ on one side:
$$
e^{x/y} y - 5y^2 = e^{x/y} x \frac{d y}{d x} - y^2 \frac{d y}{d x}
$$

Factor out $$ \frac{d y}{d x} $$:
$$
e^{x/y} y - 5y^2 = \left(e^{x/y} x - y^2\right) \frac{d y}{d x}
$$

Now, solve for $$ \frac{d y}{d x} $$:
$$
\frac{d y}{d x} = \frac{e^{x/y} y - 5y^2}{e^{x/y} x - y^2}
$$

Thus, the final result for $$ \frac{d y}{d x} $$ is:
$$
\frac{d y}{d x} = \frac{e^{x/y} y - 5y^2}{e^{x/y} x - y^2}
$$
$$ \frac{dy}{dx} = \frac{e^{x/y} \cdot y - 5y^2}{e^{x/y} \cdot x - y^2} $$

18. [-/1 Points] DETAILS MY NOTE Find \( \frac{d y}{d x} \) by implicit differentiation. \( e^{x / y}=5 x-y \) \( \frac{d y}{d x}=\square \)

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