Q. \(\displaystyle \frac{d}{dx}\left(2e^{x}\right)\)

We want to find the derivative of the function \(2e^x\).

Step 1: Identify the constant factor.

The function is \(2e^x\). The number \(2\) is a constant multiplier.

Step 2: Use the constant multiple rule.

The constant multiple rule says:

\[
\frac{d}{dx}\bigl(c \cdot f(x)\bigr) = c \cdot \frac{d}{dx}f(x)
\]

Here, \(c=2\) and \(f(x)=e^x\). So:

\[
\frac{d}{dx}\bigl(2e^x\bigr) = 2 \cdot \frac{d}{dx}\bigl(e^x\bigr)
\]

Step 3: Differentiate \(e^x\).

A key fact is:

\[
\frac{d}{dx}\bigl(e^x\bigr) = e^x
\]

Step 4: Substitute back and simplify.

So:

\[
\frac{d}{dx}\bigl(2e^x\bigr) = 2 \cdot e^x
\]

Final Answer:

\[
\frac{d}{dx}\bigl(2e^x\bigr) = 2e^x
\]