Find the implicit derivative $\frac{d}{dx}\left(\sin\left(x\right)=x\left(1+\tan\left(y\right)\right)\right)$

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$\frac{d}{dx}\left(\sin\left(x\right)\right)=\frac{d}{dx}\left(x\left(1+\tan\left(y\right)\right)\right)$

Learn how to solve problems step by step online. Find the implicit derivative d/dx(sin(x)=x(1+tan(y))). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x and g=1+\tan\left(y\right). The derivative of the linear function is equal to 1. The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if {f(x) = \sin(x)}, then {f'(x) = \cos(x)\cdot D_x(x)}.