Calculus help? What's the answer here?

f(x, y) is differentiable at the point P(4 , 4)

df(x, y)/dx at P is 8

df(x, y)/dy at P is -2

f(4, 4) = 5

In terms of x only, the situation compares to a linear approximation (near some quadratic or cubic or whatever), given by a tangent at (4, 4) with slope 8.

If that was say y = 8x + c then with x = 4,

y = 8*4 + c, but when x = 4.08, y = 8*4.08 + c

This diversion is just to emphasise that the contribution

due to the change in x, is 8 * 0.08

In a similar way, the contribution due to the change in y is -2 * -0.09

Since f started with a value of 5 at (4, 4), we estimate its value

at the point on a tangent line (4.08, 3.91) as 5 + 8*.08 + 2*.09 = 5.82

Stanschim gave this more concisely earlier.

You left out part of the problem when typing it; the original contains all information needed.

5 + 8(.08) - 2(-.09) = 5.82