What it shows
Every exponential curve y = aˣ has a neat habit: its slope at any point is a fixed multiple of its height there. For y = 2ˣ the slope is always about 0.693 times the height; for y = 3ˣ it is about 1.099 times. Somewhere between 2 and 3 there is a base where the multiple is exactly 1, so the slope equals the height everywhere. That base is e = 2.71828…, and that is why calculus prefers eˣ to every other exponential: it is its own derivative.
The page shows this three ways. The main picture draws y = aˣ with a bar for the height at P and a blue step for the rise of the tangent over 1 unit, so you can compare them by eye. The interest box finds the same e in money: 1 rupee at 100 % a year, compounded more and more often, grows to (1 + 1/n)ⁿ, which creeps up to e. And the mirror picture shows ln x, the reflection of eˣ in the line y = x, whose slope at x is 1/x.
It follows exponential and logarithmic functions in Class 11 maths and the derivatives of eˣ, aˣ and ln x in continuity and differentiability in Class 12 maths. Start from the idea of a slope in derivative slope; afterwards, the chain rule gears handles e^(3x) and ln(x² + 1), and derivative steps differentiates any mix of them with the working shown.
How to use it
- Level. Class 11 keeps to the picture and the interest, and works the slope out from a small step h. Class 12 writes the derivative rule d/dx aˣ = aˣ ln a, draws eˣ faintly for comparison and adds the ln x mirror card.
- Slide the base a from 1.2 to 4. Read Slope ÷ height: it changes with a but not with P.
- Move P with its slider or by dragging along the picture. Height and slope both change; their ratio does not.
- Press Find the base where slope = height. The base glides to 2.718 and the page names it.
- Slide n in the interest box from 1 to a million and watch the money approach e. The table shows how slowly the gap closes.
- In the mirror card, move the point on ln x and compare the two slopes.
Worked example
With a = 2 and P at x = 1, the height is 2¹ = 2. The tangent’s slope is 2 × ln 2 = 2 × 0.6931 = 1.3863, so slope ÷ height = 0.6931. Move P to x = −1: the height is 0.5, the slope 0.3466, and the ratio is still 0.6931. Now press the button: at a = e the ratio is 1, and at x = 1 the height and slope are both 2.7183.
In the interest box, n = 12 (monthly) gives (1 + 1/12)¹² = 2.613035, n = 365 gives 2.714567, and n = 1,000,000 gives 2.718280, just 0.0000014 short of e.
Tips
- ln a is just the name for the “slope ÷ height” number of aˣ. That is why ln 2 ≈ 0.693 and ln e = 1.
- Because ln x is the mirror of eˣ, at x = 2 its slope is 1/2: the mirror point on eˣ has height 2, so slope 2, and mirrored slopes flip to their reciprocals.
- Any exponential can be rewritten with e: aˣ = e^(x ln a).
FAQ
Why not just use base 10? You can, but then every derivative carries an extra factor ln 10 ≈ 2.3026. With base e that factor is 1.
Is e exactly 2.718? No, it never ends or repeats: 2.718281828459045…
Is this page part of NCERT? No. fibub is independent. It follows the topics of the Class 11 and 12 maths syllabus and uses none of the textbook’s text or exercises.
Where the numbers come from
- Standard results: d/dx aˣ = aˣ ln a, e = lim (1 + 1/n)ⁿ = Σ 1/k!, ln x as the inverse of eˣ, d/dx ln x = 1/x. Written for fibub; no textbook text, figures or exercises are used.
- Checked against mpmath (30 digits, shipped with sympy 1.14): ln 2, ln 3, ln 2.5 and ln 10, (1 + 1/n)ⁿ for n from 1 to a million, partial sums of Σ 1/k!, the value of e and slopes of 2ˣ.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.