Let z=√2x+5y

Question

. Then:

The rate of change in z
at (2,4)
as we change x
but hold y
fixed is
, and

The rate of change in z
at (2,4)
as we change y
but hold x
fixed is

Answer 1

To find the rate of change in z with respect to x, we need to take the partial derivative of z with respect to x while holding y fixed:

∂z/∂x = ∂/∂x (√2x + 5y)

Using the chain rule, we have:

∂z/∂x = ∂/∂x (√2x) = (√2)(1) = √2

So the rate of change in z with respect to x, while holding y fixed, is √2.

To find the rate of change in z with respect to y, we need to take the partial derivative of z with respect to y while holding x fixed:

∂z/∂y = ∂/∂y (√2x + 5y)

Using the chain rule, we have:

∂z/∂y = ∂/∂y (5y) = 5

So the rate of change in z with respect to y, while holding x fixed, is 5.