Question
Download Solution PDFWhat is the value of \(\frac{{{d^n}({x^m})}}{{d{x^n}}}\) for m < n, m = n, m > n?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation:
\(\frac{{{d^n}({x^m})}}{{d{x^n}}}\) = ? ; for m < n, m = n, and m > n
∵ \(\frac{{{d^n}({x^m})}}{{d{x^n}}}\) = \(m\frac{d^{n~-~1 ~}(x^{m~-~1})}{dx^{n~-~1}}~=~m(m~-~1)\frac{d^{n~-~2 ~}(x^{m~-~2})}{dx^{n~-~2}}~=~........\)
Case: I
Since m > n, hence the cycle will move up to (m - n) times, and at last
\(\frac{{{d^n}({x^m})}}{{d{x^n}}}\) = \(m(m~-~1)(m~-~2)...................[m~-~(n~-~1)]x^{m~-~n}\)
Hence, for m > n,
\(\frac{{{d^n}({x^m})}}{{d{x^n}}}\) = mPn x(m - n)........................................................(1)
Case: II
For m = n:
From equation (1),
\(\frac{{{d^n}({x^m})}}{{d{x^n}}}\) = nPn x(n - n) = n!
Case: III
For m < n:
\(\frac{{{d^n}({x^m})}}{{d{x^n}}}\) = mPn x(m - n) = 0 ( Because mPn = 0; For m < n )
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