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Answer :
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).
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