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 Video by Zeeshan Zamurred Exponentials and Logarithms

Edexcel AS Level Maths: 14.2 y = eˣWatch the full walkthrough before the notes below.

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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.

\frac{d}{d x} (e^{x}) = e^{x}

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

Differentiating e^(kx)

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

\frac{d}{d x} (e^{k x}) = k e^{k x}

Multiply by the coefficient of

x

1Bring down the 3.

Answer

3 e^{3 x}

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.

\frac{d}{d x} (e^{x}) = e^{x}

eˣ is its own derivative.

\frac{d}{d x} (e^{k x}) = k e^{k x}

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).

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