Question

For f,g and h differentiable functions, find a formula for the first order partial derivatives of each function. Explain in words how each formula is applied.

a) u(x,y)=f(x)+g(y)
b) v(x,y)=f(x)g(y)
c) w(x,y)=h(f(x)g(y)

Answer 1

The first order partial derivatives of u(x,y), v(x,y), and w(x,y) are found using basic differentiation rules, the product rule, and the chain rule respectively, which are based on how each variable affects the function while the other variable is held constant.

Step-by-step explanation:

We're looking to find the first order partial derivatives for the functions u(x,y)=f(x)+g(y), v(x,y)=f(x)g(y), and w(x,y)=h(f(x)g(y)). These derivatives are found by applying the rules of differentiation to each function.

• For the function u(x,y), the partial derivative with respect to x is given by:

∂u/∂x = f'(x)

• The partial derivative of u with respect to y is given by:

∂u/∂y = g'(y)

• For the function v(x,y), the partial derivatives are found using the product rule:

∂v/∂x = f'(x)g(y)
∂v/∂y = f(x)g'(y)

• For the function w(x,y), we need to apply the chain rule:

∂w/∂x = h'[f(x)g(y)] · f'(x)g(y)
∂w/∂y = h'[f(x)g(y)] · f(x)g'(y)

Each partial derivative is calculated by considering how the function changes as one variable varies while the other is held constant. For the product and chain rules, we differentiate one function and multiply by the other function staying constant or its derivative, respectively.