If  4sin(xy)−5cos(xy)=45, then dydx=

# If  $4\mathrm{sin}\left(xy\right)-5\mathrm{cos}\left(xy\right)={4}^{5}$, then $\frac{dy}{dx}=$

1. A

$\frac{-x}{y}$

2. B

$\frac{-y}{x}$

3. C

$\frac{4\mathrm{sin}\left(xy\right)-5\mathrm{cos}\left(xy\right)}{4\mathrm{cos}\left(xy\right)+5\mathrm{sin}\left(xy\right)}$

4. D

$\frac{4\mathrm{cos}\left(xy\right)+5\mathrm{sin}\left(xy\right)}{4\mathrm{sin}\left(xy\right)-5\mathrm{cos}\left(xy\right)}$

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### Solution:

$\begin{array}{l}4\mathrm{cos}\left(xy\right)\left[x\frac{dy}{dx}+y\left(1\right)\right]+5\mathrm{sin}\left(xy\right)\left[x.\frac{dy}{dx}+y\left(1\right)\right]=0\\ \left[x\frac{dy}{dx}+y\right]\left[4\mathrm{cos}\left(xy\right)+5\mathrm{sin}\left(xy\right)\right]=0\left[x\frac{dy}{dx}+y\right]\left[4\mathrm{cos}\left(xy\right)+5\mathrm{sin}\left(xy\right)\right]=0\\ x\frac{dy}{dx}=-y⇒\frac{dy}{dx}=-\frac{y}{x}\end{array}$

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