Consider the function f(\mu)=\sum_{i=1}(x_{i}-\mu)^{2}, where x_[i}=i,\ i=1,\ 2,\ \cdots,\ n What is the

Bobbie Comstock 2021-12-16 Answered
Consider the function
f(μ)=i=1(xiμ)2, where
xi=i, i=1, 2, , n
What is the first and second derivative os f(μ)?
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Expert Answer

Janet Young
Answered 2021-12-17 Author has 32 answers
Step 1
ddμf(μ)=2i=1n(x1μ)=2i=1nxi+2nμ
d2dμ2f(μ)=2n
In the 1st step on line 1, the chain rule is applied. The term within the brackets is differentiated, producing a -1.
Then the bracket itself is differentiated, producing the 2 at the front. The 2nd step on line 1 involves no differentiation. Instead, the bracket is split into two terms.
The second term has an n because it is simply the summation from i=1 to i=n of a constant.
The summation of a constant is equal to n multiplied by the constant. Then for the second line, there are no extra rules.
The first term becomes 0 because it's a constant and the second term loses μ.
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accimaroyalde
Answered 2021-12-18 Author has 29 answers
Step 1
Adding an answer here to further clarify the other ones which are simply answers without steps. To get the first derivative, this can be re-written as:
ddμsun(xμ)2
=ddμ(xμ)2
After that its
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nick1337
Answered 2021-12-28 Author has 575 answers

Step 1
Here we have one function (xiμ) that is part of a larger function in that it is raised to a power of 2, (μi)2
According to the extended power rule, we multiply the derivative of the outer function
(μi)2x
the derivative of the inner function (xiμ)
The derivative of the outer function brings the 2 down in front as
2×(xiμ)
and the derivative of the inner function
(xiμ) is -1.
So the -2 comes from multiplying the two derivatives according to the extend power rule:
2×(xiμ)×1=2(xiμ)
Answer:
f(μ)=2i=1n(xiμ) and f(μ)=2n

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