Final answer:
To find the first partial derivatives of the function w = 9zexyz, differentiate the function with respect to each variable (x, y, z) one at a time, treating the other variables as constants.
Step-by-step explanation:
The first partial derivative of the given function w = 9zexyz with respect to x can be found by treating y and z as constants and differentiating the function with respect to x. Similarly, the first partial derivative with respect to y can be found by treating x and z as constants and differentiating the function with respect to y. Finally, the first partial derivative with respect to z can be found by treating x and y as constants and differentiating the function with respect to z.
For example, to find the first partial derivative with respect to x, we differentiate the function 9zexyz with respect to x, treating y and z as constants:
dw/dx = 9zyzejx = 9xyzex
Similarly, the first partial derivatives with respect to y and z can be found:
dw/dy = 9xzexz
dw/dz = 9xyexy