To find the first and second order partial derivatives of the function:

$z = 4x^3y^2$

First Order Partial Derivatives

1. Partial derivative with respect to $x$:

   $\frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (4x^3y^2)$

   Using the power rule for differentiation on $x$:

   $\frac{\partial z}{\partial x} = 4 \cdot 3x^2 \cdot y^2 = 12x^2y^2$

2. Partial derivative with respect to $y$:

   $\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (4x^3y^2)$

   Using the power rule for differentiation on $y$:

   $\frac{\partial z}{\partial y} = 4x^3 \cdot 2y = 8x^3y$

Second Order Partial Derivatives

1. Second partial derivative with respect to $x$:

   $\frac{\partial^2 z}{\partial x^2} = \frac{\partial}{\partial x} \left( 12x^2y^2 \right)$

   Differentiating $12x^2y^2$ with respect to $x$:

   $\frac{\partial^2 z}{\partial x^2} = 12 \cdot 2x \cdot y^2 = 24xy^2$

2. Second partial derivative with respect to $y$:

   $\frac{\partial^2 z}{\partial y^2} = \frac{\partial}{\partial y} \left( 8x^3y \right)$

   Differentiating $8x^3y$ with respect to $y$:

   $\frac{\partial^2 z}{\partial y^2} = 8x^3 \cdot 1 = 8x^3$

3. Mixed partial derivative with respect to $x$ and $y$:

   • First, $\frac{\partial}{\partial x} \left( \frac{\partial z}{\partial y} \right)$:

     $\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial x} \left( 8x^3y \right)$

     Differentiating $8x^3y$ with respect to $x$:

     $\frac{\partial^2 z}{\partial x \partial y} = 8 \cdot 3x^2 \cdot y = 24x^2y$

   • Second, $\frac{\partial}{\partial y} \left( \frac{\partial z}{\partial x} \right)$:

     $\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial}{\partial y} \left( 12x^2y^2 \right)$

     Differentiating $12x^2y^2$ with respect to $y$:

     $\frac{\partial^2 z}{\partial y \partial x} = 12x^2 \cdot 2y = 24x^2y$

   Since mixed partial derivatives are equal, we have:

   $\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} = 24x^2y$

Summary of the Partial Derivatives

• First Order:

  • $\frac{\partial z}{\partial x} = 12x^2y^2$
  • $\frac{\partial z}{\partial y} = 8x^3y$

• Second Order:

  • $\frac{\partial^2 z}{\partial x^2} = 24xy^2$
  • $\frac{\partial^2 z}{\partial y^2} = 8x^3$
  • $\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} = 24x^2y$

Would you like more details on any of these steps, or have further questions?

Related Questions:

1. What are higher-order partial derivatives and their applications?
2. How do we interpret mixed partial derivatives geometrically?
3. Can we find the gradient and Hessian matrix of $z = 4x^3y^2$?
4. How do mixed partial derivatives relate to Clairaut’s theorem?
5. What are some practical uses of partial derivatives in real-world problems?

Tip:

When dealing with partial derivatives, always check if mixed derivatives are equal; this verifies smoothness and continuity in most contexts.