Final answer:
The partial derivatives of the function z = f(x)g(y) are ∆z/∆x = f'(x)g(y) and ∆z/∆y = f(x)g'(y), where you differentiate each part while treating the other variable as a constant.
Step-by-step explanation:
The question asks for the partial derivatives of the function z = f(x)g(y) with respect to x and y. In calculus, when taking the partial derivative of a function with two variables where each is a function of a single variable, we treat the other variable as a constant. Applying this to the given function:
- For ∆z/∆x, hold y constant and differentiate f(x) with respect to x. The result is f'(x)g(y) because g(y) acts as a constant.
- For ∆z/∆y, hold x constant and differentiate g(y) with respect to y. The result is f(x)g'(y) because f(x) acts as a constant.
Therefore, ∆z/∆x = f'(x)g(y) and ∆z/∆y = f(x)g'(y).