To differentiate \( e^{x^2 + y^2 - xy} \), we need to apply the **chain rule**. The expression is a composite function where \( u = x^2 + y^2 - xy \) is the exponent of \( e^u \). We'll compute the partial derivatives with respect to both \( x \) and \( y \).

### 1. Derivative with respect to \( x \):

\[
\frac{\partial}{\partial x} \left( e^{x^2 + y^2 - xy} \right) = e^{x^2 + y^2 - xy} \cdot \frac{\partial}{\partial x} (x^2 + y^2 - xy)
\]

Now, compute the partial derivative of \( u = x^2 + y^2 - xy \) with respect to \( x \):

\[
\frac{\partial}{\partial x} (x^2 + y^2 - xy) = 2x - y
\]

Thus, the partial derivative with respect to \( x \) is:

\[
\frac{\partial}{\partial x} \left( e^{x^2 + y^2 - xy} \right) = e^{x^2 + y^2 - xy} \cdot (2x - y)
\]

### 2. Derivative with respect to \( y \):

\[
\frac{\partial}{\partial y} \left( e^{x^2 + y^2 - xy} \right) = e^{x^2 + y^2 - xy} \cdot \frac{\partial}{\partial y} (x^2 + y^2 - xy)
\]

Now, compute the partial derivative of \( u = x^2 + y^2 - xy \) with respect to \( y \):

\[
\frac{\partial}{\partial y} (x^2 + y^2 - xy) = 2y - x
\]

Thus, the partial derivative with respect to \( y \) is:

\[
\frac{\partial}{\partial y} \left( e^{x^2 + y^2 - xy} \right) = e^{x^2 + y^2 - xy} \cdot (2y - x)
\]

### Conclusion:
The partial derivatives are:
- With respect to \( x \): \( e^{x^2 + y^2 - xy} \cdot (2x - y) \)
- With respect to \( y \): \( e^{x^2 + y^2 - xy} \cdot (2y - x) \)

Would you like further details or have any questions about this process?

### Related questions:
1. How do you apply the chain rule in multivariable calculus?
2. What is the derivative of \( e^{x^2} \) with respect to \( x \)?
3. How do you find the total derivative of a function involving both \( x \) and \( y \)?
4. What is the difference between partial and total derivatives?
5. Can you explain implicit differentiation with respect to \( y \) in a multivariable function?

### Tip: 
For exponential functions involving composite expressions, always remember to use the chain rule, which includes differentiating the exponent first and then multiplying by the derivative of the exponent.

Learn how to find the partial derivatives of e^(x^2 + y^2 - xy) with respect to x and y by applying the chain rule. This solution explains the key steps and concepts, including partial differentiation and the chain rule, suitable for college-level calculus students.

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