Q. \[ \frac{d}{dx}\left(4e^{x}\right) \]

Q: What is the derivative of 4e^x?

Use the constant multiple rule: \frac{d}{dx}(4e^x)=4\frac{d}{dx}(e^x)=4e^x.

Q: Why is \frac{d}{dx}(e^x)=e^x?

The derivative of the exponential base e satisfies \frac{d}{dx}(e^x)=e^x by definition and key limit properties.

Q: What is the derivative of k e^x for a constant k?

\frac{d}{dx}(k e^x)=k\frac{d}{dx}(e^x)=k e^x.

Q: What is the derivative of 4e^{3x}?

Apply the chain rule: \frac{d}{dx}(4e^{3x})=4\cdot e^{3x}\cdot 3=12e^{3x}.

Q: What is the derivative of 4e^{x+2}?

Rewrite e^{x+2}=e^2 e^x: \frac{d}{dx}(4e^{x+2})=4e^{x+2} (since \frac{d}{dx}(e^{x+2})=e^{x+2}).

Q: How do you differentiate 4e^x+7?

Differentiate term-by-term: \frac{d}{dx}(4e^x+7)=4e^x+0=4e^x.

Q: What is the derivative of \frac{1}{4}e^x?

Constant multiple rule again: \frac{d}{dx}\left(\frac{1}{4}e^x\right)=\frac{1}{4}e^x.

Answer

Use the rule \(\frac{d}{dx}\left(e^x\right)=e^x\). Since \(4\) is a constant, it stays outside.

\[
\frac{d}{dx}\left(4e^x\right)=4e^x
\]

Detailed Explanation

We want to find the derivative of \(4e^x\) with respect to \(x\), meaning we compute \(\dfrac{d}{dx}\left(4e^x\right)\).

Step 1: Identify constants and functions

In the expression \(4e^x\), the number \(4\) is a constant multiplier, and \(e^x\) is the variable part.

Step 2: Use the constant multiple rule

The constant multiple rule says:

\[
\frac{d}{dx}\left(c \cdot f(x)\right) = c \cdot \frac{d}{dx}\left(f(x)\right)
\]

Here, \(c = 4\) and \(f(x) = e^x\).

So:

\[
\frac{d}{dx}\left(4e^x\right) = 4 \cdot \frac{d}{dx}\left(e^x\right)
\]

Step 3: Use the derivative of \(e^x\)

A key fact is:

\[
\frac{d}{dx}\left(e^x\right) = e^x
\]

Step 4: Combine the results

Substitute \( \frac{d}{dx}\left(e^x\right) = e^x \) into the expression:

\[
\frac{d}{dx}\left(4e^x\right) = 4e^x
\]

Final Answer

\[
\frac{d}{dx}\left(4e^x\right) = 4e^x
\]

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Calculus FAQ

What is the derivative of \(4e^x\)?

Use the constant multiple rule: \(\frac{d}{dx}(4e^x)=4\frac{d}{dx}(e^x)=4e^x\).

Why is \(\frac{d}{dx}(e^x)=e^x\)?

The derivative of the exponential base \(e\) satisfies \(\frac{d}{dx}(e^x)=e^x\) by definition and key limit properties.

What is the derivative of \(k e^x\) for a constant \(k\)?

\(\frac{d}{dx}(k e^x)=k\frac{d}{dx}(e^x)=k e^x\).

What is the derivative of \(4e^{3x}\)?

Apply the chain rule: \(\frac{d}{dx}(4e^{3x})=4\cdot e^{3x}\cdot 3=12e^{3x}\).

What is the derivative of \(4e^{x+2}\)?

Rewrite \(e^{x+2}=e^2 e^x\): \(\frac{d}{dx}(4e^{x+2})=4e^{x+2}\) (since \(\frac{d}{dx}(e^{x+2})=e^{x+2}\)).

How do you differentiate \(4e^x+7\)?

Differentiate term-by-term: \(\frac{d}{dx}(4e^x+7)=4e^x+0=4e^x\).

What is the derivative of \(\frac{1}{4}e^x\)?

Constant multiple rule again: \(\frac{d}{dx}\left(\frac{1}{4}e^x\right)=\frac{1}{4}e^x\).

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