Q. Find the derivative of \(6e^x\).

Q: What is the derivative of 6e^x?

Use the constant multiple rule: \frac{d}{dx}(6e^x)=6\frac{d}{dx}(e^x)=6e^x.

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

e^x is its own derivative by definition/construction: \frac{d}{dx}(e^x)=e^x.

Q: Does the constant 6 affect the derivative?

Yes, but only by scaling: \frac{d}{dx}(6f(x))=6f'(x). So \frac{d}{dx}(6e^x)=6e^x.

Q: What if it is 6e^{2x}?

Chain rule: \frac{d}{dx}(6e^{2x})=6\cdot e^{2x}\cdot 2=12e^{2x}.

Q: What is the derivative of 6e^x+3?

Differentiate termwise: \frac{d}{dx}(6e^x+3)=6e^x+0=6e^x.

Q: What is the derivative of e^x/6?

Constant factor rule: \frac{d}{dx}\left(\frac{e^x}{6}\right)=\frac{1}{6}\frac{d}{dx}(e^x)=\frac{1}{6}e^x.

Answer

To differentiate \(6e^x\), use the fact that \(\frac{d}{dx}(e^x)=e^x\) and the constant rule.

\[
\frac{d}{dx}\left(6e^x\right)=6\cdot \frac{d}{dx}\left(e^x\right)=6e^x
\]

Detailed Explanation

We want to find the derivative of the function

\[ f(x) = 6e^x. \]

Step 1: Identify the constant and the function.

The number \(6\) is a constant multiplier, and \(e^x\) is the exponential part of the function.

Step 2: Use the constant multiple rule for derivatives.

If \(f(x) = c \cdot g(x)\), then

\[ f'(x) = c \cdot g'(x). \]

Here, \(c = 6\) and \(g(x) = e^x\).

Step 3: Use the derivative rule for the exponential function.

A key fact is that

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

Step 4: Combine the steps.

Differentiate \(6e^x\) by keeping the \(6\) and differentiating \(e^x\):

\[ \frac{d}{dx}\left(6e^x\right) = 6 \cdot \frac{d}{dx}\left(e^x\right) = 6e^x. \]

Final Answer:

\[ \frac{d}{dx}\left(6e^x\right) = 6e^x. \]

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Calculus FAQ

What is the derivative of \(6e^x\)?

Use the constant multiple rule: \(\frac{d}{dx}(6e^x)=6\frac{d}{dx}(e^x)=6e^x\).

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

\(e^x\) is its own derivative by definition/construction: \(\frac{d}{dx}(e^x)=e^x\).

Does the constant \(6\) affect the derivative?

Yes, but only by scaling: \(\frac{d}{dx}(6f(x))=6f'(x)\). So \(\frac{d}{dx}(6e^x)=6e^x\).

What if it is \(6e^{2x}\)?

Chain rule: \(\frac{d}{dx}(6e^{2x})=6\cdot e^{2x}\cdot 2=12e^{2x}\).

What is the derivative of \(6e^x+3\)?

Differentiate termwise: \(\frac{d}{dx}(6e^x+3)=6e^x+0=6e^x\).

What is the derivative of \(e^x/6\)?

Constant factor rule: \(\frac{d}{dx}\left(\frac{e^x}{6}\right)=\frac{1}{6}\frac{d}{dx}(e^x)=\frac{1}{6}e^x\).

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