Final answer:
The partial derivative of the function f(x, y) = 100x⁰.⁷⁴y⁰.²⁶with respect to x is 74 * x⁻⁰.²⁶* y⁰.²⁶, and with respect to y is 26 * x⁰.⁷⁴ * y⁻⁰.⁷⁴. To find the value of ∂f/∂x at a specific point (x, y), substitute the respective x and y values into 74 * x⁻⁰.²⁶* y^0.26.
Step-by-step explanation:
The student has asked to find the partial derivatives of the function f(x, y) = 100x⁰.⁷⁴y⁰.²⁶with respect to each variable and then to evaluate the partial derivative with respect to x at the point (x, y).
To find the partial derivative of f with respect to x, denoted as ∂f/∂x, we treat y as a constant and differentiate f with respect to x using the power rule:
∂f/∂x = 100 * 0.74 * x⁰.⁷⁴-1y0.26 = 74 * x⁰.²⁶y⁰.²⁶
To evaluate ∂f/∂x at any point (x, y), we simply substitute the given values for x and y into the equation:
∂f/∂x | (x, y) = 74 * x^-0.26y^0.26
Partial Derivative with respect to y
To find the partial derivative of f with respect to y, denoted as ∂f/∂y, we treat x as a constant and differentiate f with respect to y:
∂f/∂y = 100 * x⁰.⁷⁴*0.26*y^0.26-1 = 26 * x⁰.⁷⁴*y⁻⁰.⁷⁴