Q. \(\frac{d}{dx}\left(5e^x\right)\)

Q: Why does the constant 5 stay outside the derivative?

Because \frac{d}{dx}(k f(x))=k f'(x) for constant k. So \frac{d}{dx}(5e^x)=5\frac{d}{dx}(e^x)=5e^x

Q: What is the derivative of 5e^{2x}?

Use chain rule: \frac{d}{dx}\left(5e^{2x}\right)=5\cdot e^{2x}\cdot 2=10e^{2x}

Q: What is the derivative of 5e^{x}+e^{x}?

Combine coefficients first: 5e^x+e^x=6e^x. Then \frac{d}{dx}(6e^x)=6e^x

Answer

To differentiate \(5e^x\), use the rule that the derivative of \(e^x\) is \(e^x\), and constants stay in front.

\[
\frac{d}{dx}\left(5e^x\right)=5\frac{d}{dx}\left(e^x\right)=5e^x
\]

Detailed Explanation

We want to find the derivative of the function \(5e^x\).

Step 1: Identify the constant and the function to differentiate

The function is \(5e^x\). Here, \(5\) is a constant multiplier, and \(e^x\) is the exponential part we need to differentiate.

Step 2: Use the constant multiple rule

The constant multiple rule says that if \(f(x)=c\cdot g(x)\), then

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

So, with \(c=5\) and \(g(x)=e^x\), we get

\[\frac{d}{dx}\left(5e^x\right)=5\cdot \frac{d}{dx}\left(e^x\right).\]

Step 3: Use the derivative of the exponential function

A key fact is that

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

Step 4: Combine the results

Substitute \(\frac{d}{dx}\left(e^x\right)=e^x\) into the expression from Step 2:

\[\frac{d}{dx}\left(5e^x\right)=5\cdot e^x.\]

Final Answer

\[\boxed{5e^x}\]

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Calculus FAQ

What is the derivative of \(5e^x\)?

\( \frac{d}{dx}\left(5e^x\right)=5e^x \)

Why does the constant \(5\) stay outside the derivative?

Because \( \frac{d}{dx}(k f(x))=k f'(x) \) for constant \(k\). So \( \frac{d}{dx}(5e^x)=5\frac{d}{dx}(e^x)=5e^x \)

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

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

What is the derivative of \(7e^x-3\)?

\( \frac{d}{dx}\left(7e^x-3\right)=7e^x-0=7e^x \)

What is the derivative of \(5e^{2x}\)?

Use chain rule: \( \frac{d}{dx}\left(5e^{2x}\right)=5\cdot e^{2x}\cdot 2=10e^{2x} \)

What is the derivative of \(5e^{x}+e^{x}\)?

Combine coefficients first: \(5e^x+e^x=6e^x\). Then \( \frac{d}{dx}(6e^x)=6e^x \)

Solve the derivative of 5e^x.
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