Find the partial derivatives of the following functions: 4x^2 y^2+5x^2+2y+9 z=(5x^3+y)/(2x+3y) z=(3xy-x^2 )^8

**For the function $f(x,y) = 4x^2 y^2 + 5x^2 + 2y + 9$:**

To find the partial derivatives, we take the derivative with respect to each variable separately.

1. Partial derivative with respect to $x$:
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(4x^2 y^2 + 5x^2 + 2y + 9)
\frac{\partial f}{\partial x} = 8xy^2 + 10x

2. Partial derivative with respect to $y$:
\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(4x^2 y^2 + 5x^2 + 2y + 9)
\frac{\partial f}{\partial y} = 8x^2y + 2

**For the function $z=\frac{5x^3+y}{2x+3y}$:**

1. Partial derivative with respect to $x$:
\frac{\partial z}{\partial x} = \frac{\partial}{\partial x}\left(\frac{5x^3+y}{2x+3y}\right)
\frac{\partial z}{\partial x} = \frac{15x^2(2x+3y) - (5x^3 + y)\cdot 2}{(2x+3y)^2}
\frac{\partial z}{\partial x} = \frac{30x^3 + 45xy - 10x^3 - 2y}{(2x+3y)^2}
\frac{\partial z}{\partial x} = \frac{20x^3 + 45xy - 2y}{(2x+3y)^2}

2. Partial derivative with respect to $y$:
\frac{\partial z}{\partial y} = \frac{\partial}{\partial y}\left(\frac{5x^3+y}{2x+3y}\right)
\frac{\partial z}{\partial y} = \frac{5x^3 + 2x+3y}{(2x+3y)^2} - \frac{(5x^3+y)\cdot 3}{(2x+3y)^2}
\frac{\partial z}{\partial y} = \frac{5x^3 + 2x+3y - 15x^3 - 3y}{(2x+3y)^2}
\frac{\partial z}{\partial y} = \frac{-10x^3 + 2x}{(2x+3y)^2}

**For the function $z=(3xy-x^2)^8$:**

1. Partial derivative with respect to $x$:
\frac{\partial z}{\partial x} = \frac{\partial}{\partial x}((3xy-x^2)^8)
\frac{\partial z}{\partial x} = 8(3xy-x^2)^7(3y-2x)

2. Partial derivative with respect to $y$:
\frac{\partial z}{\partial y} = \frac{\partial}{\partial y}((3xy-x^2)^8)
\frac{\partial z}{\partial y} = 8(3xy-x^2)^7(3x)

**Answer:**
1. $\frac{\partial f}{\partial x} = 8xy^2 + 10x$
2. $\frac{\partial f}{\partial y} = 8x^2y + 2$
3. $\frac{\partial z}{\partial x} = \frac{20x^3 + 45xy - 2y}{(2x+3y)^2}$
4. $\frac{\partial z}{\partial y} = \frac{-10x^3 + 2x}{(2x+3y)^2}$
5. $\frac{\partial z}{\partial x} = 8(3xy-x^2)^7(3y-2x)$
6. $\frac{\partial z}{\partial y} = 8(3xy-x^2)^7(3x)$