Question

Use differentials to approximate the change in z for the given change in the independent variables. If z = ln(x²y), what is the approximate change in z when x increases by dx and y increases by dy?

Answer 1

To approximate the change in z when z = ln(x²y), differentiate to get dz = 2(dx/x) + (dy/y), which can be used to estimate the change given small increments in x and y.

Step-by-step explanation:

To approximate the change in z for the given change in the independent variables when z = ln(x²y), we use differentials. We start by differentiating z with respect to x and y:

dz = (d/dx)ln(x²y) * dx + (d/dy)ln(x²y) * dy

Using the properties of logarithms, we rewrite z as ln(x²) + ln(y), then differentiate:

dz = [2*(1/x)dx] + (1/y)dy

Thus, the approximate change in z when x increases by dx and y increases by dy is:

dz = 2(dx/x) + (dy/y)

This formula allows us to calculate the approximate change in z given small changes dx and dy in the variables x and y, respectively.