14.2PureCore

y = eˣ

Among all exponential functions, one is special: $y = e^x$ , where $e \approx 2.718$ . It is the natural exponential, and its defining magic is that it is its own derivative — the reason it appears everywhere in calculus.

25 min Exponentials and Logarithms

Edexcel AS Level Maths: 14.2 y = eˣCurated by Kepler, not produced by us. Watch it before the notes below.

Open on YouTube

What you'll be able to do

Know the value and significance of e
Recognise the graph of y = eˣ
Use the result that the derivative of eˣ is eˣ
Differentiate y = e^(kx)

The number e

$e \approx 2.71828$ is an irrational constant, the natural base for exponentials. The function $y = e^x$ behaves like any $y = a^x$ (through $(0,1)$ , asymptote $y = 0$ ) but with a unique calculus property.

Its own derivative

The gradient of $y = e^x$ at any point equals the $y$ -value there. In other words, differentiating $e^x$ gives $e^x$ back — unchanged.

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

The only function (up to a constant) that is its own derivative.

Differentiating e^(kx)

For $y = e^{kx}$ , differentiating brings the constant $k$ down as a multiplier.

\dfrac{d}{dx}\!\left(e^{kx}\right) = k\,e^{kx}

Multiply by the coefficient of

x

Example 1Differentiate

y = e^{3x}

1Bring down the 3.

Answer

3e^{3x}

Tip — e^(kx) differentiates to k·e^(kx) — the e-part never changes, you just pull k out front.

Formula recap

e \approx 2.718

The natural base.

\dfrac{d}{dx}(e^{x}) = e^{x}

eˣ is its own derivative.

\dfrac{d}{dx}(e^{kx}) = k e^{kx}

Chain factor k.

Common mistakes to avoid

Differentiating eˣ using the power rule.

The power rule is for xⁿ; d/dx(eˣ) = eˣ.

Forgetting the k when differentiating e^(kx).

d/dx(e^(kx)) = k·e^(kx).

Key takeaways

e ≈ 2.718 is the natural exponential base.
y = eˣ is its own derivative.
d/dx(e^(kx)) = k·e^(kx).

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 8 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.