Final answer:
To find the partial derivatives ∂z/∂x and ∂z/∂y for the equation e^z = 3xyz, we perform implicit differentiation and solve for the derivatives, which involves applying exponential function properties and logarithms.
Step-by-step explanation:
The question asks for the partial derivatives of z with respect to x and y from the equation ez = 3xyz. To find these derivatives, we need to implicitly differentiate both sides of the equation with respect to each variable while treating the other variable as a constant.
To find ∂z/∂x, differentiate with respect to x, getting zez ∂z/∂x = 3yz + 3xy(∂z/∂x), which we can solve for ∂z/∂x after rearranging. Similarly, to find ∂z/∂y, differentiate with respect to y, resulting in zez ∂z/∂y = 3xz + 3xy(∂z/∂y), and then solve for ∂z/∂y.
Once we have the derivatives, we can substitute the values of x, y, and z if needed to find the numerical values of these partial derivatives. The differentiation here involves implicit differentiation and requires an understanding of exponential functions along with the properties of logarithms when isolating z.