Example 3

Find the partial derivatives of

f(x,y,z) = sin(x²y - z)

at the point (1,0, 0).

To find f_x(x, y, z) we treat y and z as constants.

f_x(x,y,z) = 2xy cos(x²y - z).

f_y(x,y,z) = x² cos(x²y - z).

f_z(x,y,z) = -cos(x²y - z).

Thus

f_x(1,0,0) = 2 · 1 · 0 cos(1²·0 - 0) = 0.

f_y(1,0,0) = 1² cos(1² · 0 - 0) = 1.

f_z(1,0,0) = -cos(1² · 0 - 0) = -1.