Question

Differentitate with respect to x $\frac{e^{x}sinx}{{(x^2+2)}^3}$.

Answer 1

We have,

Let $y=\frac{e^x\sin x}{{(x^2+2)}^3}$

$y=e^x\sin x{(x^2+2)}^{-3}$

Applying formula

$\frac{d}{dx}(I.II.III)=I.II\frac{d}{dx}III+I.III\frac{d}{dx}II+II.III\frac{d}{dx}I$

Now,

$\frac{dy}{dx}=e^x\sin x\frac{d}{dx}{(x^2+2)}^{-3}+e^x(x^2+2)\frac{d}{dx}\sin x^{-3}+\sin x(x^2+2)\frac{d}{dx}{e}^x$

$=e^x\sin x(-3){(x^2+2)}^{-4}\left(2x\right)+e^x(x^2+2)\cos x+\sin x(x^2+2)e^x$

$=-6x{e}^x\sin x{(x^2+2)}^{-4}+e^x(x^2+2)\cos x+\sin x(x^2+2)e^x$

Hence, this is the answer.