if ( y=arctan left(e^{2 x}\right) ), then ( \frac{d y}{d x}= )

Explanation:

Step1: Let (u = e^{2x})

The function (y=\arctan(u)). By the chain - rule, (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). The derivative of (y = \arctan(u)) with respect to (u) is (\frac{dy}{du}=\frac{1}{1 + u^{2}}).

Step2: Find (\frac{du}{dx})

Since (u = e^{2x}), using the chain - rule for (y = e^{v}) where (v = 2x). The derivative of (e^{v}) with respect to (v) is (e^{v}), and the derivative of (v=2x) with respect to (x) is (2). So (\frac{du}{dx}=e^{2x}\cdot2 = 2e^{2x}).

Step3: Substitute (u) back and calculate (\frac{dy}{dx})

Substitute (u = e^{2x}) into (\frac{dy}{du}), we get (\frac{dy}{du}=\frac{1}{1+(e^{2x})^{2}}=\frac{1}{1 + e^{4x}}). Then (\frac{dy}{dx}=\frac{1}{1 + e^{4x}}\cdot2e^{2x}=\frac{2e^{2x}}{1 + e^{4x}}).

Answer:

(\frac{2e^{2x}}{1 + e^{4x}})