Find the first partial derivatives of the function.
f(x,y,z) = xe^(y/z)
a. fx = ?
b. fy = ?
c. fz = ?
To find the first partial derivatives of the function f(x, y, z) = xe^(y/z), we differentiate the function with respect to each variable while treating the other variables as constant.
a. To find fx, we differentiate f(x, y, z) with respect to x and treat y and z as constants.
∂f/∂x = e^(y/z) (By differentiating x, we get 1, and since e^(y/z) is treated as a constant)
b. To find fy, we differentiate f(x, y, z) with respect to y and treat x and z as constants.
∂f/∂y = x * (e^(y/z)) * (-z^(-2)) (By applying the chain rule, we differentiate e^(y/z) which becomes e^(y/z) * (1/z) and then differentiate y/z which becomes (-1/z^2))
c. To find fz, we differentiate f(x, y, z) with respect to z and treat x and y as constants.
∂f/∂z = x * (e^(y/z)) * (-y/z^2) (By applying the chain rule, we differentiate e^(y/z) which becomes e^(y/z) * (1/z) and then differentiate y/z which becomes (-y/z^2))
Therefore:
a. fx = e^(y/z)
b. fy = x * (e^(y/z)) * (-z^(-2))
c. fz = x * (e^(y/z)) * (-y/z^2)