$\frac{d^{n}}{d x^{n}} (e^{2 x} + e^{- 2 x}) =$

$(e^{2 x} + {(- 1)}^{n} e^{- 2 x})$

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$2^{n} (e^{2 x} + e^{- 2 x})$

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$2^{n} (e^{2 x} + {(- 1)}^{n} e^{- 2 x})$

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None of these

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Solution

The correct option is C
$2^{n} (e^{2 x} + {(- 1)}^{n} e^{- 2 x})$

Explanation for the correct options:
Let $y = (e^{2 x} + e^{- 2 x})$
Differentiate w.r.t. $x$
$\frac{d y}{d x} = 2 e^{2 x} - 2 e^{- 2 x}$
Again differentiate w.r.t. $x$
$\frac{d^{2} y}{d x^{2}} = 4 e^{2 x} + 4 e^{- 2 x}$
$= 2^{2} e^{2 x} + {(- 2)}^{2} e^{- 2 x}$
Similarly, $\frac{d^{n} y}{d x^{n}} =$ $2^{n} (e^{2 x} + {(- 1)}^{n} e^{- 2 x})$
Hence Option(C) is correct.

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