Question

find the derivative. y = √(1 + e^{2x})

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt{1 + e^{2x}}$ as $y=(1 + e^{2x})^{\frac{1}{2}}$.

Step2: Apply the chain - rule

Let $u = 1+e^{2x}$, then $y = u^{\frac{1}{2}}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$: $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}=\frac{1}{2(1 + e^{2x})^{\frac{1}{2}}}$.

Step3: Find $\frac{du}{dx}$

Since $u = 1+e^{2x}$, then $\frac{du}{dx}=2e^{2x}$ (using the rule that the derivative of $e^{ax}$ is $ae^{ax}$ and the derivative of a constant is 0).

Step4: Calculate $\frac{dy}{dx}$

Multiply $\frac{dy}{du}$ and $\frac{du}{dx}$: $\frac{dy}{dx}=\frac{1}{2(1 + e^{2x})^{\frac{1}{2}}}\cdot2e^{2x}=\frac{e^{2x}}{\sqrt{1 + e^{2x}}}$.

Answer:

$\frac{e^{2x}}{\sqrt{1 + e^{2x}}}$