Example 1

Find the partial derivatives of the function

f(x, y) = x² + 3xy - 8y

at the point (2, -1).

To find f_x(x, y), we treat y as a constant,

f_x(x,y) = 2x+3y.

To find f_y(x, y), we treat x as a constant,

f_y(x,y) = 3x - 8.

Thus

f_x(2, -1) = 2 · 2 + 3(-l) = 1, f_y(2, -1) = 3 · 2 - 8 = -2.

Figure 11.3.2 shows the surface z = f(x, y) and the tangent lines at the point (2,-1).

Figure 11.3.2