-1 = xy - tan(x-y)−1=xy−tan(x−y)
frac{d}{dx}(-1) = frac{d}{dx}(xy - tan(x-y))ddx(−1)=ddx(xy−tan(x−y))
0 = frac{d}{dx}(xy) - frac{d}{dx}(tan(x-y))0=ddx(xy)−ddx(tan(x−y))
= xfrac{d}{dx}(y) + yfrac{d}{dx}(x) - sec^2(x-y)frac{d}{dx}(x-y)=xddx(y)+yddx(x)−sec2(x−y)ddx(x−y)
= xfrac{dy}{dx} + y - sec^2(x-y)(1-frac{dy}{dx})=xdydx+y−sec2(x−y)(1−dydx)
Now we just have to "shift terms" to make frac{dy}{dx}dydx the subject of the formula.
sec^2(x-y) - y = (sec^2(x-y) + x)frac{dy}{dx}sec2(x−y)−y=(sec2(x−y)+x)dydx
frac{dy}{dx} = frac{sec^2(x-y) - y}{sec^2(x-y) + x}dydx=sec2(x−y)−ysec2(x−y)+x