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

Q: What is the derivative of e^x?

Since e^x equals its own derivative: \frac{d}{dx}\left(e^x\right)=e^x .

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

By the definition of e^x as the unique function whose derivative equals itself: \frac{d}{dx}\left(e^x\right)=e^x .

Q: What is the derivative of e^{u} where u=u(x)?

Use the chain rule: \frac{d}{dx}\left(e^{u(x)}\right)=e^{u(x)}\cdot u'(x) .

Q: What is \frac{d}{dx}\left(e^{ax}\right) ?

Let u=ax. Then u'=a: \frac{d}{dx}\left(e^{ax}\right)=a e^{ax} .

Q: What is \frac{d}{dx}\left(3e^x\right) ?

Constants factor out: \frac{d}{dx}\left(3e^x\right)=3e^x .

Q: What is \frac{d}{dx}\left(e^{-x}\right) ?

Chain rule with u=-x, so u'=-1: \frac{d}{dx}\left(e^{-x}\right)=-e^{-x} .

Answer

The function \(e^x\) means the derivative has the same form. Using the fact that \(\frac{d}{dx}\left(e^x\right)=e^x\), we get:

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

Detailed Explanation

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

Step 1: Recall a key derivative rule.

There is a fundamental fact from calculus: the exponential function with base \(e\) is its own derivative.

That rule is:

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

This means that when you differentiate \(e^x\), you get \(e^x\) back again.

Step 2: Apply the rule to the given function.

The given function is exactly \(e^x\), so we directly substitute into the rule.

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

Final Answer:

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

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Calculus FAQ

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

Since \(e^x\) equals its own derivative: \( \frac{d}{dx}\left(e^x\right)=e^x \).

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

By the definition of \(e^x\) as the unique function whose derivative equals itself: \( \frac{d}{dx}\left(e^x\right)=e^x \).

What is the derivative of \(e^{u}\) where \(u=u(x)\)?

Use the chain rule: \( \frac{d}{dx}\left(e^{u(x)}\right)=e^{u(x)}\cdot u'(x) \).

What is \( \frac{d}{dx}\left(e^{ax}\right) \)?

Let \(u=ax\). Then \(u'=a\): \( \frac{d}{dx}\left(e^{ax}\right)=a e^{ax} \).

What is \( \frac{d}{dx}\left(3e^x\right) \)?

Constants factor out: \( \frac{d}{dx}\left(3e^x\right)=3e^x \).

What is \( \frac{d}{dx}\left(e^{-x}\right) \)?

Chain rule with \(u=-x\), so \(u'=-1\): \( \frac{d}{dx}\left(e^{-x}\right)=-e^{-x} \).

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