Question

Find the total differential of z = f(x, y), where f(x, y) = 5sin(4x^y). What's the total differential, dz?

Answer 1

To find the total differential dz of the function z = f(x, y) = 5sin(4x^y), we can use the chain rule and find the partial derivatives with respect to x and y. Then, we multiply the partial derivatives by the differentials dx and dy to obtain the total differential dz.

Step-by-step explanation:

The total differential of a function can be found by taking the partial derivatives of the function with respect to each variable and then multiplying them by the corresponding differentials. In this case, the function is z = f(x, y) = 5sin(4x^y). To find the total differential dz, we can use the chain rule.

First, find the partial derivatives: ∂f/∂x = 20yx^(y-1)cos(4x^y) and ∂f/∂y = 20x^ysin(4x^y)ln(x).

Then, multiply the partial derivatives by the differentials: dz = ∂f/∂x * dx + ∂f/∂y * dy.