Final Answer
$-2\cos\left(x\right)\sin\left(x\right)+\frac{\left(-4xy^{\prime}\sin\left(y\right)^{3}\cos\left(y\right)-\sin\left(y\right)^{4}\right)\sin\left(x\right)^2+2x\sin\left(y\right)^{4}\sin\left(x\right)\cos\left(x\right)}{\sin\left(x\right)^{4}}=-2\tan\left(\frac{\pi}{2}-x\right)\sec\left(\frac{\pi}{2}-x\right)^2\tan\left(\frac{\pi}{2}-y\right)^2-2y^{\prime}\tan\left(\frac{\pi}{2}-x\right)^2\tan\left(\frac{\pi}{2}-y\right)\sec\left(\frac{\pi}{2}-y\right)^2$
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Step-by-step Solution
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Simplifying
$\frac{d}{dx}\left(\cos\left(x\right)^2+x\frac{-\sin\left(y\right)^2}{\sin\left(x\right)^2}\sin\left(y\right)^2=\tan\left(\frac{\pi}{2}-x\right)^2\tan\left(\frac{\pi}{2}-y\right)^2-1\right)$
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$\frac{d}{dx}\left(\cos\left(x\right)^2+x\frac{-\sin\left(y\right)^2}{\sin\left(x\right)^2}\sin\left(y\right)^2=\tan\left(\frac{\pi}{2}-x\right)^2\tan\left(\frac{\pi}{2}-y\right)^2-1\right)$
Learn how to solve problems step by step online. Find the implicit derivative of cos(x)^2+(-sin(y)^2)/(sin(x)^2)xsin(y)^2=tan(pi/2-x)^2tan(pi/2-y)^2-1. Simplifying. Simplifying. Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. The derivative of a sum of two or more functions is the sum of the derivatives of each function.
Final Answer
$-2\cos\left(x\right)\sin\left(x\right)+\frac{\left(-4xy^{\prime}\sin\left(y\right)^{3}\cos\left(y\right)-\sin\left(y\right)^{4}\right)\sin\left(x\right)^2+2x\sin\left(y\right)^{4}\sin\left(x\right)\cos\left(x\right)}{\sin\left(x\right)^{4}}=-2\tan\left(\frac{\pi}{2}-x\right)\sec\left(\frac{\pi}{2}-x\right)^2\tan\left(\frac{\pi}{2}-y\right)^2-2y^{\prime}\tan\left(\frac{\pi}{2}-x\right)^2\tan\left(\frac{\pi}{2}-y\right)\sec\left(\frac{\pi}{2}-y\right)^2$