Exercise with differential of a function

[Felafel]

Homework Statement

Let $Q(x)=\sum_{j,i=1}^nC_{ij}x^ix^j$
and $C_{ij}=C_{ji}; Q(x)>0 \forall x\neq 0$
$f(x)=[Q(x)]^{\frac{p}{2}}$ find df(x)

The Attempt at a Solution

$Q(x)=\begin{matrix} (c_{11} & c_{12}... & c_{1n}) \\ (... & ... & ...) \\ (c_{n1} & c_{n2} & c_{nn}) \end{matrix}^{p/2}$multplied the column $(x_1,...x_n)^{p/2}$
$df(x)=\frac{p}{2} \sum_{i,j=1}^n(C_{ji})^{\frac{p}{2}} \cdot x^{\frac{p-2}{2}} dx$
$\forall x_i$ with $i=1,...n$
is it correct?
thank you in advance

 

[mighty2000]

Thank you for your question. Your attempt at a solution is partially correct. Here are a few points to consider:

1. When taking the derivative of $f(x)$, you should use the chain rule. This means that you should multiply the derivative of the outer function $[Q(x)]^{\frac{p}{2}}$ by the derivative of the inner function $Q(x)$. In this case, the inner function is a sum, so you will need to use the sum rule as well.

2. Your calculation for $Q(x)$ is not entirely clear. It looks like you are trying to write out the matrix representation of $Q(x)$, but it is not clear how you are using the indices $i$ and $j$. Remember that the sum in the definition of $Q(x)$ is over all possible values of $i$ and $j$, so you will need to use these indices in your calculation of $Q'(x)$.

3. It is also not clear what you mean by "multiplied the column $(x_1,...x_n)^{p/2}$". Remember that in the definition of $Q(x)$, the variables $x_1,x_2,...,x_n$ are raised to different powers depending on the values of $i$ and $j$. So it is not clear how you are raising the entire column to a single power.

4. Finally, your answer should be a function of $x$, not just a sum over the indices $i$ and $j$. Remember that the derivative of a function is a new function that tells you the slope of the original function at each point. So your answer should be a function that depends on $x$.

I hope this helps you to revise your solution. Good luck!