Exponential derivatives

Derivative of exp(x): Why eˣ Is Its Own Derivative

The derivative of e^x is e^x. The notation exp(x) means the same exponential function and is used in the calculator examples below. For exp(g(x)), preserve the original exponent and multiply by its derivative; other positive constant bases also contribute ln(a).

Direct answer

ddx[ex]=ex\frac{\mathrm{d}}{\mathrm{d}x}\left[e^{x}\right] = e^{x}

For a real aˣ, a>0; exp(g(x)) also requires g′(x).

When to use derivative of e^x / exp(x)

Choose the method from the top-level mathematical structure before manipulating symbols. Conditions are part of the task, not optional notes.

Ask first 1

Which structure controls the method?

What to do

Identify the base, then decide whether the exponent is a composite function.

Ask first 2

Which condition can change the answer?

What to do

For a real aˣ, a>0; exp(g(x)) also requires g′(x).

Ask first 3

How should the result be checked?

What to do

Divide the result by the original exponential; the remaining factor should be the exponent derivative.

Factor the difference quotient

Factor exp(x) from [exp(x+h)−exp(x)]/h.

The remaining limit [exp(h)−1]/h equals one by the defining property of e.

Thus exp′(x)=exp(x); for aˣ, write exp(x ln a).

For exp(g(x)), preserve g(x) and multiply by g′(x).

1
ex+h=ex ehe^{x + h} = e^{x}\,e^{h}
2
lim [exp(h)−1]/h=1
3
[exp(g(x))]'=exp(g(x))g'(x)

Worked examples

ddx[ex]\frac{\mathrm{d}}{\mathrm{d}x}\left[e^{x}\right]
exe^{x}

The basic exponential rule.

ddx[e4 x]\frac{\mathrm{d}}{\mathrm{d}x}\left[e^{4\,x}\right]
4 e4 x4\,e^{4\,x}

The inner derivative is 4.

ddx[2x]\frac{\mathrm{d}}{\mathrm{d}x}\left[2^{x}\right]
2^x ln(2)

A non-e base contributes ln(2).

Common mistakes

  • • Treating every base like e.
  • • Replacing the exponent by its derivative.
  • • Forgetting g′(x).
  • • Adding ln(a) when the base is e.

Error clinic: locate the first invalid step

Incorrect attempt

[exp(x²)]'=exp(2x)

Why it fails

The exponent was replaced instead of preserved.

Correction

[exp(x²)]'=2x exp(x²)

Incorrect attempt

[2^x]'=2^x

Why it fails

Only base e has unit logarithmic factor.

Correction

[2^x]'=2^x ln(2)

Practice the same idea with a changed structure

Name the rule and its conditions before revealing the reference answer. Explain every sign and factor.

ddx[e4 x]\frac{\mathrm{d}}{\mathrm{d}x}\left[e^{4\,x}\right]

Check before solving

The inner derivative is 4.

Reveal the reference answer

Reference answer

4 e4 x4\,e^{4\,x}
ddx[2x]\frac{\mathrm{d}}{\mathrm{d}x}\left[2^{x}\right]

Check before solving

A non-e base contributes ln(2).

Reveal the reference answer

Reference answer

2^x ln(2)

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

Derivatives:exe^{x}Derivatives:e4 xe^{4\,x}
Open the calculator without a preset: Derivatives

Frequently asked questions

Are e^x and exp(x) the same function?+

Yes. Both mean the exponential function with base e. The example buttons use exp(x); its derivative is exp(x).

What is the derivative of aˣ?+

For a>0, it is aˣ ln(a).

What is d/dx exp(x²)?+

It is 2x exp(x²), not exp(2x).

Conclusion

exp(x) is unchanged by differentiation; composite exponents add the inner derivative and other bases add ln(a).

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