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I'm supposed to implicitly differentiate $\sin(x+y)=2x-2y$. I've already taken the first derivative and got

$$ \left(\frac{dy}{dx}+1\right)\cdot\cos(y+x)=-2\left(\frac{dy}{dx}-1\right) $$

www.derivative-calculator.net says solving this equation for $\frac{dy}{dx}$ equals

$$ \frac{dy}{dx}=-\frac{\cos(y+x)-2}{\cos(y+x)+2} $$

But I'm lost when it comes to the algebra used to rewrite the equation in terms of $\frac{dy}{dx}$. Any help getting from point A to B will be greatly appreciated.

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  • $\begingroup$ These 2 equations are trivially equal. $\endgroup$ – crbah Mar 8 '16 at 14:50
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$$ \left(\frac{dy}{dx}+1\right)\cdot\cos(y+x)=-2\left(\frac{dy}{dx}-1\right) $$ $$\frac{dy}{dx}\cos(y+x)+\cos(y+x)=-2\frac{dy}{dx}+2$$ $$\frac{dy}{dx}\cos(y+x)+2\frac{dy}{dx}=2-\cos(y+x)$$ $$\frac{dy}{dx}(\cos(y+x)+2)=-(-2+\cos(y+x))$$ $$\frac{dy}{dx}=-\frac{\cos(y+x)-2}{\cos(y+x)+2}$$

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So, $$\sin(x+y) = 2(x-y)$$ thus $$\cos(x+y)+\frac{dy}{dx}\cos(x+y)=2-2\frac{dy}{dx}$$ Or $$\cos(x+y)\left(1+\frac{dy}{dx}\right)=2\left(1-\frac{dy}{dx}\right)$$ Re-arranging $$\cos(x+y)\frac{dy}{dx}+2\frac{dy}{dx}=2-\cos(x+y)$$ Hence, $$\frac{dy}{dx}(2+\cos(x+y) = -(\cos(x+y)-2)$$ From which $$\frac{dy}{dx} = -\frac{\cos(x+y)-2}{\cos(x+y)+2}$$

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