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Question
If \[f\left( x \right) = \left( \frac{x^l}{x^m} \right)^{l + m} \left( \frac{x^m}{x^n} \right)^{m + n} \left( \frac{x^n}{x^l} \right)^{n + 1}\] the f' (x) is equal to _____________ .
Options
1
0
\[x^{l + m + n}\]
none of these
Solution
`0`
We have,
\[ \Rightarrow f\left( x \right) = x^{l^2 - m^2} \times x^{m^2 - n^2} \times x^{n^2 - l^2} \]
\[ \Rightarrow f\left( x \right) = x^\left( l^2 - m^2 + m^2 - n^2 + n^2 - l^2 \right) \]
\[ \Rightarrow f\left( x \right) = x^0 \]
\[ \Rightarrow f\left( x \right) = 1\]
\[ \Rightarrow f'\left( x \right) = 0\]
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