Consider the function f(μ)=∑i=1(xi−μ)2, where xi=i, i=1, 2, ⋯, n What is the first and second derivative os f(μ)?

Name: Consider the function f(μ)=∑i=1(xi−μ)2, where xi=i, i=1, 2, ⋯, n What is the first and second derivative os f(μ)?
Uploaded: 2022-02-23T17:22:57+00:00
Description: Consider the function f(μ)=∑i=1(xi−μ)2, where xi=i, i=1, 2, ⋯, n What is the first and second derivative os f(μ)?

Question

Consider the function  f(μ)=∑i=1(xi−μ)2, where  xi=i, i=1, 2, ⋯, n  What is the first and second derivative os f(μ)?

Janet Young · Accepted Answer

Step 1ddμf(μ)=−2∑i=1n(x1−μ)=−2∑i=1nxi+2nμd2dμ2f(μ)=2nIn 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&#039;s a constant and the second term loses μ.

accimaroyalde · Accepted Answer

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(x−μ)2  =∑ddμ(x−μ)2  After that its

nick1337 · Accepted Answer

Step 1Here we have one function (xi−μ) that is part of a larger function in that it is raised to a power of 2, (μ−i)2According to the extended power rule, we multiply the derivative of the outer function(μ−i)2xthe derivative of the inner function (xi−μ)The derivative of the outer function brings the 2 down in front as2×(xi−μ)and the derivative of the inner function(xi−μ) is -1.So the -2 comes from multiplying the two derivatives according to the extend power rule:2×(xi−μ)×−1=−2(xi−μ)Answer:f′(μ)=−2∑i=1n(xi−μ) and f″(μ)=2n

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

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