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Differentiation in Rn
Differentiation in Rn
3 questions
Difficulty 3-5
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For a function
f
:
R
n
→
R
, partial derivatives
∂
f
/
∂
x
i
measure one-variable rates of change. How is this different from the total derivative (gradient)?
Hide and think first
A.
Partial derivatives capture directional rates along axes; the gradient assembles them into a vector that captures all first-order info
B.
Partial derivatives always exist when the gradient exists, but the gradient can exist without the partials
C.
Partial derivatives and the gradient are synonymous; they refer to the same mathematical object
D.
The gradient is a scalar summary of the partials, equal to
∑
i
∂
f
/
∂
x
i
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