MathematicsIf 4sin(xy)−5cos(xy)=45, then dydx=

If $4 \sin (x y) - 5 \cos (x y) = 4^{5}$ , then $\frac{d y}{d x} =$

A
$\frac{- x}{y}$
B
$\frac{- y}{x}$
C
$\frac{4 \sin (x y) - 5 \cos (x y)}{4 \cos (x y) + 5 \sin (x y)}$
D
$\frac{4 \cos (x y) + 5 \sin (x y)}{4 \sin (x y) - 5 \cos (x y)}$

Solution:

$4 \cos (x y) [x \frac{d y}{d x} + y (1)] + 5 \sin (x y) [x . \frac{d y}{d x} + y (1)] = 0$

$[x \frac{d y}{d x} + y] [4 \cos (x y) + 5 \sin (x y)] = 0$

$x \frac{d y}{d x} = - y \Rightarrow \frac{d y}{d x} = - \frac{y}{x}$

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