Certainly! To find the derivative of [tex]\(x^2 \log(x)\)[/tex] with respect to [tex]\(Y\)[/tex], let's go through the steps:
1. Identify the function and the variables: The function given is [tex]\(x^2 \log(x)\)[/tex], and we need to differentiate this function with respect to the variable [tex]\(Y\)[/tex].
2. Understand the relationship between the variables: In this expression, the independent variable [tex]\(x\)[/tex] and the logarithm term [tex]\(\log(x)\)[/tex] don't involve the variable [tex]\(Y\)[/tex]. This means that [tex]\(x\)[/tex] and [tex]\(Y\)[/tex] are considered independent, and any partial derivative of a function with respect to a variable that does not appear in the function directly will be zero.
3. Concept of Partial Derivatives: A partial derivative of a function of multiple variables with respect to one of those variables measures how the function changes as that particular variable changes, while keeping the others constant.
4. Apply the concept to the given function: Since [tex]\(x^2 \log(x)\)[/tex] does not contain [tex]\(Y\)[/tex], the function remains constant with respect to changes in [tex]\(Y\)[/tex].
Therefore, the derivative of [tex]\(x^2 \log(x)\)[/tex] with respect to [tex]\(Y\)[/tex] is:
[tex]\[
\frac{\partial}{\partial Y} (x^2 \log(x)) = 0
\][/tex]
So, the result is [tex]\(0\)[/tex].
