Derivative of a square root with exponential function

Question

So I have the following function:

$f(x)= \sqrt{e^{2x}}$

After applying the chain rule I sit with:

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

From there I got:

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

While the apparent correct answer is $e^{x}$

If anyone could help explain where I went wrong, or what I am missing I would greatly appreciate it.

Answer 1

Hint: $\sqrt{e^{2x}} = e^{\left(\dfrac{2x}{2}\right)}$

Answer 2

Hint: $$\sqrt{e^{2x}} = \sqrt{e^{x}e^{x}} = e^x$$

Although you still did get the correct answer of $\frac{e^{2x}}{\sqrt{e^{2x}}}$. With a little algebra you would have found that $$\frac{e^{2x}}{\sqrt{e^{2x}}}=e^x$$

Answer 3

Hint:

$$e^{2x}=e^x \times e^x$$

Answer 4

$\sqrt{e^{2x}} = (e^{2x})^{\frac{1}{2}} = e^{x}$

Which has derivative $e^x$