Find the differential of the function [tex]w = x^6 \sin(y^2 z)[/tex].

(dw = _, dx + _, dy + ___, dz)

A. [tex](6x^5 \sin(y^2 z) \, dx + 2x^6 yz \, dy + x^6 y^2 \, dz)[/tex]
B. [tex](6x^5 \sin(y^2 z) \, dx + x^6 y^2 \, dz + 2x^6 yz \, dy)[/tex]
C. [tex](6x^5 \sin(y^2 z) \, dx + x^6 yz \, dy + x^6 y^2 \, dz)[/tex]
D. [tex](6x^5 \sin(y^2 z) \, dx + x^6 y^2 \, dz + x^6 yz \, dy)[/tex]

Question

High School

Answer :

ramprinces279 ramprinces279 · Answer 1 · 2025-01-24T16:25:09-05:00

The differential of the function (w=x⁶sin(y²z)) is dw = 6x⁵sin(y²z), dx + 2x⁶yz, dy + x⁶y², dz. Option a) is the correct answer.

Given w = x⁶sin(y²z), differentiate with respect to each variable:

dw = (∂w/∂x)dx + (∂w/∂y)dy + (∂w/∂z)dz

Find the total differential dw by taking the partial derivatives with respect to x, y, and z.

Calculate the partial derivatives:

∂w/∂x = 6x⁵sin(y²z)

∂w/∂y = 2x⁶yz

∂w/∂z = x⁶y²

Write the total differential dw as a sum of the partial differentials:

dw = (∂w/∂x)dx + (∂w/∂y)dy + (∂w/∂z)dz

Substitute the partial derivatives into the differential expression:

dw = 6x⁵sin(y²z)dx + 2x⁶yzdy + x⁶y²dz

Therefore, the differential of the function (w=x⁶sin(y²z)) is dw = 6x⁵sin(y²z)dx + 2x⁶yzdy + x⁶y²dz, which corresponds to Option a).