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Okay this may sound stupid but i need a little help.what do $\\large \\frac{d}{dx}$ and $\\large \\frac{dy}{dx}$ mean

I need a thorough explanation Rankeya has given a valid answer to the written question, but i realize now i was too vague Secondly, i looked up the correct exercise in jacobson and found that the following exercise is precisely to show that it does hold for all division rings Stupid gut feelings.i'm accepting this answer and reposting the correct question.

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Eigenvectors corresponding to distinct eigenvalues of a symmetric matrix must be orthogonal to each other Eigenvectors corresponding to the same eigenvalue need not be orthogonal to each other However, since every subspace has an orthonormal basis, you can find orthonormal bases for each eigenspace, so you can find an orthonormal basis of eigenvectors. Is this a valid notation for real $m \\times n$ matrices

$m$ and $n$ are known If it is not, then what would be the right notation for the set. According to symbolic matlab and wolframalpha, $\\frac{\\partial x(t)}{\\partial x} = 0, \\frac{\\partial x}{\\partial x} = 1$ i came across this while trying to. I don't know how to evaluate it

I know there is one method using the gamma function

But i want to know the solution using a calculus method like polar coordinates

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