Find 
dy
dx
 using partial derivatives.
exy + y^ ey = 6

To find \( \frac{dy}{dx} \) using partial derivatives for the given equation:

\[
e^{xy} + y e^{y} = 6
\]

we will implicitly differentiate both sides of the equation with respect to \( x \), treating \( y \) as a function of \( x \) (i.e., \( y = y(x) \)).

### Step 1: Differentiate the first term \( e^{xy} \)
Using the chain rule:
\[
\frac{d}{dx} \left( e^{xy} \right) = e^{xy} \cdot \frac{d}{dx}(xy) = e^{xy} \left( y + x \frac{dy}{dx} \right)
\]
(where \( \frac{dy}{dx} \) is the derivative of \( y \) with respect to \( x \)).

### Step 2: Differentiate the second term \( y e^y \)
This is a product of two functions, so we apply

Find dy/dx using partial derivatives.
e^xy + y^ ey = 6

Learn how to find dy/dx using implicit differentiation for the equation e^(xy) + y * e^y = 6. This problem covers advanced calculus concepts, including partial derivatives and the chain rule, and is suitable for undergraduate students.

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