Question

1. find the derivative.

y = \sqrt{1 + e^{2x}}
y =

Explanation:

Step1: Rewrite the function

Let $u = 1 + e^{2x}$, then $y=\sqrt{u}=u^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$. Using the power rule, if $y = u^{\frac{1}{2}}$, then $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}=\frac{1}{2\sqrt{u}}$.

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

Since $u = 1+e^{2x}$, then $\frac{du}{dx}=2e^{2x}$ (using the sum rule and the chain - rule for $e^{2x}$: if $v = 2x$, then $\frac{d(e^{v})}{dv}=e^{v}$ and $\frac{dv}{dx}=2$, so $\frac{d(e^{2x})}{dx}=e^{2x}\cdot2$).

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

Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula: $\frac{dy}{dx}=\frac{1}{2\sqrt{u}}\cdot2e^{2x}$. Replace $u$ with $1 + e^{2x}$, we get $\frac{dy}{dx}=\frac{e^{2x}}{\sqrt{1 + e^{2x}}}$.

Answer:

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