Find the differential of the function w=xsin(2yz^4).

Question

Find the differential of the function w=xsin(2yz^4).

dw= ? dx + ? dy +? dz

Answer 1

For a differentiable function of three variables , w = ƒ (x, y, z ) , we define the differentials dx, dy, dz, to be

independent variables ; that is they can be given any values .

Then the differential dz , also called the total differential, is defined by

dw = ƒ_x(x,y,z) dx + ƒ_y(x,y,z) dy +ƒ_z(x,y,z) dz = (∂ƒ /∂x) dx + (∂ƒ /∂y) dy + (∂ƒ /∂z) dz .

∂ƒ /∂x = sin(2z⁴y)

∂ƒ /∂y =2xz⁴ cos(2z⁴y)

∂ƒ /∂z = 8xyz³ cos(2z⁴y)

Thus dw = (∂ƒ /∂x) dx + (∂ƒ /∂y) dy + (∂ƒ /∂z) dz .=

dw = sin(2z⁴y) d x + 2xz⁴ cos(2z⁴y) d y + 8xyz³ cos(2z⁴y) d z