Find 

f Subscript xx​, 

f Subscript xy​,

 Subscript  

f Subscript yx​, 

and 

f Subscript yy

 for the following function.​ (Remember, 

f Subscript yx 

means to differentiate with respect to y and then with respect to​ x.) 

​f(x, 

​y)equals=e Superscript 15 xye15xy




Given the function \( f(x, y) = e^{15xy} \), we need to find the following partial derivatives:

1. \( f_{xx} \) (the second partial derivative of \( f \) with respect to \( x \)),
2. \( f_{xy} \) (the partial derivative of \( f \) first with respect to \( x \) and then with respect to \( y \)),
3. \( f_{yx} \) (the partial derivative of \( f \) first with respect to \( y \) and then with respect to \( x \)), and
4. \( f_{yy} \) (the second partial derivative of \( f \) with respect to \( y \)).

### 1. First-Order Partial Derivatives

#### \( f_x \) (the partial derivative of \( f \) with respect to \( x \)):
\[
f_x = \frac{\partial}{\partial x} \left( e^{15xy} \right)
\]
Using the chain rule:
\[
f_x = e^{15xy} \cdot \frac{\partial}{\partial x} (15xy)
\]
\[
f_x = e^{15xy} \cdot 15y = 15y e^{15xy}
\]

#### \( f_y \) (the partial derivative of \( f \) with respect to \( y \)):
\[
f_y = \frac{\partial}{\partial y} \left( e^{15xy} \right)
\]
Using the chain rule:
\[
f_y = e^{15xy} \cdot \frac{\partial}{\partial y} (15xy)
\]
\[
f_y = e^{15xy} \cdot 15x = 15x e^{15xy}
\]

### 2. Second-Order Partial Derivatives

#### \( f_{xx} \) (the second partial derivative of \( f \) with respect to \( x \)):
\[
f_{xx} = \frac{\partial}{\partial x} \left( 15y e^{15xy} \right)
\]
Using the product rule and chain rule:
\[
f_{xx} = 15y \cdot \frac{\partial}{\partial x} \left( e^{15xy} \right)
\]
\[
f_{xx} = 15y \cdot 15y e^{15xy} = 225y^2 e^{15xy}
\]

#### \( f_{xy} \) (the partial derivative of \( f_x \) with respect to \( y \)):
\[
f_{xy} = \frac{\partial}{\partial y} \left( 15y e^{15xy} \right)
\]
Using the product rule and chain rule:
\[
f_{xy} = 15 \cdot e^{15xy} + 15y \cdot \frac{\partial}{\partial y} \left( e^{15xy} \right)
\]
\[
f_{xy} = 15 e^{15xy} + 15y \cdot 15x e^{15xy} = 15 e^{15xy} + 225xy e^{15xy}
\]

#### \( f_{yx} \) (the partial derivative of \( f_y \) with respect to \( x \)):
\[
f_{yx} = \frac{\partial}{\partial x} \left( 15x e^{15xy} \right)
\]
Using the product rule and chain rule:
\[
f_{yx} = 15 \cdot e^{15xy} + 15x \cdot \frac{\partial}{\partial x} \left( e^{15xy} \right)
\]
\[
f_{yx} = 15 e^{15xy} + 15x \cdot 15y e^{15xy} = 15 e^{15xy} + 225xy e^{15xy}
\]
(Note that \( f_{xy} = f_{yx} \), which is expected due to Clairaut's theorem for mixed partial derivatives.)

#### \( f_{yy} \) (the second partial derivative of \( f \) with respect to \( y \)):
\[
f_{yy} = \frac{\partial}{\partial y} \left( 15x e^{15xy} \right)
\]
Using the product rule and chain rule:
\[
f_{yy} = 15x \cdot \frac{\partial}{\partial y} \left( e^{15xy} \right)
\]
\[
f_{yy} = 15x \cdot 15x e^{15xy} = 225x^2 e^{15xy}
\]

### Summary of Results

- \( f_{xx} = 225y^2 e^{15xy} \)
- \( f_{xy} = 15 e^{15xy} + 225xy e^{15xy} \)
- \( f_{yx} = 15 e^{15xy} + 225xy e^{15xy} \)
- \( f_{yy} = 225x^2 e^{15xy} \)

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How does the exponential function \( e^{15xy} \) affect the behavior of the partial derivatives compared to a polynomial function?
2. What can you infer about the function \( f(x, y) \) based on the second-order partial derivatives?
3. How would the partial derivatives change if the exponent were a different function, such as \( e^{15x^2y} \)?
4. Can you determine the Hessian matrix for the function \( f(x, y) = e^{15xy} \)?
5. How do second-order partial derivatives relate to the curvature of the surface defined by \( f(x, y) = e^{15xy} \)?

**Tip:** The mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) are equal if the function is continuously differentiable, as per Clairaut's theorem.

Explore the partial derivatives of the function f(x, y) = e^(15xy). Calculate f_xx, f_xy, f_yx, and f_yy step-by-step using the chain rule and product rule. Understand the impact of exponential functions on partial derivatives. Suitable for advanced undergraduate level.

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