"Find the gradient of the given function: z = xe^(y). Assume the variables are restricted to a domain on which the function is defined."
Thanks for the help!
To find the gradient of the given function, we need to calculate the partial derivatives of the function with respect to each variable, x and y.
First, let's find the partial derivative with respect to x (denoted as ∂z/∂x):
To do this, we treat y as a constant and differentiate the function z = xe^(y) with respect to x. The derivative of xe^(y) with respect to x is simply e^(y), since differentiating x gives 1, and e^(y) is treated as a constant.
So, ∂z/∂x = e^(y).
Next, let's find the partial derivative with respect to y (denoted as ∂z/∂y):
To do this, we treat x as a constant and differentiate the function z = xe^(y) with respect to y. The derivative of xe^(y) with respect to y can be found by applying the chain rule. The derivative of e^(y) is e^(y), and multiplying it by x gives xe^(y).
Therefore, ∂z/∂y = xe^(y).
The gradient of the function is a vector that contains both partial derivatives:
Gradient (∇z) = (∂z/∂x, ∂z/∂y) = (e^(y), xe^(y)).
Remember that the gradient gives us the direction of steepest increase of the function, and its magnitude represents the rate of change.