Q. \[ \frac{d}{dx}\left(3e^x\right) \]

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

\frac{d}{dx}\left(3e^{x}\right)=3e^{x} since \frac{d}{dx}(e^{x})=e^{x}.

Q: How do you use the constant multiple rule for derivatives?

If y=cf(x), then \frac{dy}{dx}=c\frac{df}{dx}. Here c=3, so \frac{d}{dx}(3e^{x})=3\frac{d}{dx}(e^{x})=3e^{x}.

Q: What is the derivative of e^{x}, and why?

\frac{d}{dx}(e^{x})=e^{x}. The exponential e^{x} is defined/characterized by having derivative equal to itself.

Q: What is the derivative of ae^{bx} in general?

For y=ae^{bx}, \frac{d}{dx}(ae^{bx})=ab e^{bx}.

Q: If y=3e^{x}, what is y' using basic rules?

Differentiate term-by-term: \frac{d}{dx}(3e^{x})=3\frac{d}{dx}(e^{x})=3e^{x}.

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

y'=3e^{x} and y''=3e^{x} because differentiating e^{x} again gives e^{x}.

Answer

To differentiate \(3e^x\), use the fact that \(\dfrac{d}{dx}\left(e^x\right)=e^x\). The constant \(3\) stays outside.

\[
\frac{d}{dx}\left(3e^x\right)=3e^x
\]

Detailed Explanation

We want to find the derivative of the function \(3e^x\).

Step 1: Identify the function and constants

The function is \(f(x)=3e^x\). Here, \(3\) is a constant multiplier, and \(e^x\) is the variable part.

Step 2: Use the Constant Multiple Rule

The Constant Multiple Rule says: if \(f(x)=c\cdot g(x)\), then \(f'(x)=c\cdot g'(x)\), where \(c\) is a constant.

So here, \(c=3\) and \(g(x)=e^x\). Therefore,

\[
\frac{d}{dx}\left(3e^x\right)=3\frac{d}{dx}\left(e^x\right).
\]

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 from Step 2:

\[
\frac{d}{dx}\left(3e^x\right)=3e^x.
\]

Final Answer

The derivative of \(3e^x\) is \(3e^x\).

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Calculus FAQ

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

\(\frac{d}{dx}\left(3e^{x}\right)=3e^{x}\) since \(\frac{d}{dx}(e^{x})=e^{x}\).

How do you use the constant multiple rule for derivatives?

If \(y=cf(x)\), then \(\frac{dy}{dx}=c\frac{df}{dx}\). Here \(c=3\), so \(\frac{d}{dx}(3e^{x})=3\frac{d}{dx}(e^{x})=3e^{x}\).

What is the derivative of \(e^{x}\), and why?

\(\frac{d}{dx}(e^{x})=e^{x}\). The exponential \(e^{x}\) is defined/characterized by having derivative equal to itself.

What is the derivative of \(ae^{bx}\) in general?

For \(y=ae^{bx}\), \(\frac{d}{dx}(ae^{bx})=ab e^{bx}\).

If \(y=3e^{x}\), what is \(y'\) using basic rules?

Differentiate term-by-term: \(\frac{d}{dx}(3e^{x})=3\frac{d}{dx}(e^{x})=3e^{x}\).

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

\(y'=3e^{x}\) and \(y''=3e^{x}\) because differentiating \(e^{x}\) again gives \(e^{x}\).

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