Question

find dy/dx with implicit differentiation.

(x^2)(y^2)+ xsiny = 7

Answer 1

\[ \frac{d}{dx}(x^2y^2) = 2xy^2 + x^22y\frac{dy}{dx}\]
Differentiate all like this first.

Answer 2

\[dy/dx=(-2x*y^2-\sin(y))/(x^2*2y+x*\cos(y))\]\[(x^2)\prime*y^2+x^2*(y^2)\prime*dy/dx+(x)\prime*\sin(y)+x*(\sin(y))\prime*dy/dx=0\]\[x^2*y^2+x*\sin(y)=7\]

Answer 3

Lets see if I can solve this at this effete stage:

\[ x^2y^2+ x\sin y = 7 \] Differentiating both sides w.r.t \(x\) :\[ 2xy^2+x^22y \frac {dy}{dx} + \sin y + x\cos y \frac {dy}{dx} = 0 \] \[ \implies (2xy^2 + \sin y)+ \frac {dy}{dx} \left(x^22y+x\cos y \right) = 0 \] \[ \implies \frac {dy}{dx} = -\frac{(2xy^2 + \sin y)}{(x^22y+x\cos y )} \]