use implicit differentiation to find \\frac{dy}{dx}.\nx = \\tan y

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (x) and (\tan y) with respect to (x). $$\frac{d}{dx}(x)=\frac{d}{dx}(\tan y)$$ Since (\frac{d}{dx}(x) = 1), and by the chain - rule (\frac{d}{dx}(\tan y)=\sec^{2}y\frac{dy}{dx}), we have (1=\sec^{2}y\frac{dy}{dx}).

Step2: Solve for (\frac{dy}{dx})

Recall the trigonometric identity (\sec^{2}y = 1+\tan^{2}y), and since (x = \tan y), then (\sec^{2}y=1 + x^{2}). From (1=\sec^{2}y\frac{dy}{dx}), we can solve for (\frac{dy}{dx}) by dividing both sides by (\sec^{2}y). So (\frac{dy}{dx}=\frac{1}{\sec^{2}y}). Substituting (\sec^{2}y = 1 + x^{2}), we get (\frac{dy}{dx}=\frac{1}{1 + x^{2}}).

Answer:

(\frac{dy}{dx}=\frac{1}{1 + x^{2}})