If $y = x^{4} e^{2 x}$ then $y_{10} (0)$ is equal to

2^{10}

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315 \times 2^{10}

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195 \times 2^{10}

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315 \times 2^{8}

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Solution

The correct option is C $315 \times 2^{10}$
$y = x^{4} e^{2 x} u = e^{2 x} a n d v = x^{4} y_{10} = u_{10} v + {10 C}_{1} u_{9} v_{1} + {10 C}_{2} u_{8} v_{2} + . . . . . . . . + {10 C}_{9} u v_{9} + {10 C}_{10} u v_{10}$
except $v_{4} = 4! = 24$ all other derivatives of v are zero at x = 0 i.e
$v (0) = v_{1} (0) = . . . . . . . v_{n} (0) = 0 \Rightarrow y_{10} = {10 C}_{4} u_{6} v_{4} u_{n} = 2^{n} e^{2 x} u_{6} = 2^{6} e^{2 x} u_{6} (0) = 2^{6} y_{10} (0) = {10 C}_{4} \times 24 \times 2^{6}$
$y_{10} (0) = \frac{10 \times 9 \times 8 \times 7}{2 \times 3 \times 4} \times 24 \times 2^{6} = 315 \times 2^{10}$

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