use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\\( x = \\tan y \\)\n\\( \\frac { d y…

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (x) with respect to (x) gives (1). For (\tan y), using the chain - rule ((\tan u)^\prime=\sec^{2}u\cdot u^\prime) (where (u = y) and (u^\prime=\frac{dy}{dx})), we have (\frac{d}{dx}(\tan y)=\sec^{2}y\frac{dy}{dx}). So the equation becomes (1=\sec^{2}y\frac{dy}{dx}).

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

Since (\sec^{2}y = 1+\tan^{2}y) and (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{1}{1 + x^{2}})