Question

A cola vending company estimates that their cola will be consumed by people at a rate given by: C'(t) = 9 + 0.1e^(0.06t) where C'(t) is in billions of gallons per hour. Find the total differential of z = f(x, y), where f(x, y) = 5sin(4x^y). dz = ?

Answer 1

The total differential of the function f(x, y) = 5sin(4x^y) is found by calculating its partial derivatives with respect to x and y, and then combining them with the differentials dx and dy.

Step-by-step explanation:

The question asks for the total differential of z for the function f(x, y) = 5sin(4x^y). The total differential (dz) of a function with two variables is found by partially differentiating the function with respect to each variable and then multiplying by the differentials of the variables (dx and dy).

To find dz, we need to calculate the partial derivatives of f with respect to x and y:

df/dx = 5y(4x^(y-1))cos(4x^y)

df/dy = 5log(4x)sin(4x^y)

Thus, the total differential, dz, is:

dz = (5y(4x^(y-1))cos(4x^y))dx + (5log(4x)sin(4x^y))dy