Question
Find $\frac{d y}{d x}, y=\frac{2 x+3}{3 x-2}$
✓
Answer
$ y=\frac{2 x+3}{3 x-2}$
$\text { Take } u=2 x+3 \text { and } v=3 x-2 .$
$ \therefore \frac{d u}{d x}=2 \text { and } \frac{d v}{d x}=3 $
Now, $y=\frac{u}{v}$
$ \therefore \frac{d y}{d x} =\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^{2}}$
$ =\frac{(3 x-2)(2)-(2 x+3)(3)}{(3 x-2)^{2}}$
$ =\frac{(6 x-4)-(6 x+9)}{(3 x-2)^{2}}$
$ =\frac{6 x-4-6 x-9}{(3 x-2)^{2}}$
$ =\frac{-13}{(3 x-2)^{2}} $
$\text { Take } u=2 x+3 \text { and } v=3 x-2 .$
$ \therefore \frac{d u}{d x}=2 \text { and } \frac{d v}{d x}=3 $
Now, $y=\frac{u}{v}$
$ \therefore \frac{d y}{d x} =\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^{2}}$
$ =\frac{(3 x-2)(2)-(2 x+3)(3)}{(3 x-2)^{2}}$
$ =\frac{(6 x-4)-(6 x+9)}{(3 x-2)^{2}}$
$ =\frac{6 x-4-6 x-9}{(3 x-2)^{2}}$
$ =\frac{-13}{(3 x-2)^{2}} $
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