Q101 / 160 · TS EAMCET 2006 · Mathematics

$f(x)=e^x \sin x$, then $f^{(6)}(x)$ is equal to :

  • A
    $e^{6 x} \sin 6 x$
  • Correct answer
    B
    $-8 e^x \cos x$
  • C
    $8 e^x \sin x$
  • D
    $8 e^x \cos x$

Solution

$f(x)=e^x \sin x$ $\therefore f^{\prime}(x)=e^x \cos x+\sin x e^x$ $\Rightarrow f^{\prime \prime}(x)=e^x \cos x-e^x \sin x+e^x \sin x$ $+e^x \cos x$ $=2 e^x \cos x$ Now, $\quad f^{\prime \prime \prime}(x)=-2 e^x \sin x+2 e^x \cos x$ and $\quad f^{i v}(x)=-2 e^x \sin x-2 e^x \cos x$ $+2 e^x \cos x-2 e^x \sin x$ $=-4 e^x \sin x$ $f^v(x)=-4 e^x \cos x-4 e^x \sin x$ $f^{v i}(x)=-4 e^x \cos x+4 e^x \sin x+4 e^x \sin x$$-4 e^x \cos x$ $=-8 e^x \cos x$

Hence, the correct answer is option: 'B'.

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