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Exponential & Log Derivatives

d/dx(eˣ) = eˣ

📖 All Three Exponential & Log Derivatives

These are the three core derivatives involving exponential and logarithmic functions.

Function Derivative Condition
eˣ eˣ Always
aˣ aˣ · ln a a > 0, a ≠ 1
ln x 1/x x > 0

🔤 Why d/dx(eˣ) = eˣ?

The number e ≈ 2.71828 is defined precisely so that its exponential function is its own derivative. This makes eˣ the most important function in all of calculus.

e = lim(n→∞) (1 + 1/n)ⁿ ≈ 2.71828...

📝 Example 1 — Exponential

Differentiate: y = 3eˣ + 5 · 2ˣ

1
Differentiate 3eˣ (constant multiple rule)d/dx(3eˣ) = 3eˣ
2
Differentiate 5 · 2ˣ (using aˣ rule, a=2)d/dx(5 · 2ˣ) = 5 · 2ˣ · ln 2
✓
Answer: dy/dx = 3eˣ + 5 · 2ˣ · ln 2

📝 Example 2 — Logarithm

Differentiate: y = x² · ln x (using product rule)

1
Identify f and g for product rulef = x² → f' = 2x    g = ln x → g' = 1/x
2
Apply: fg' + gf'x²·(1/x) + ln x·2x
3
Simplifyx + 2x ln x
✓
Answer: dy/dx = x(1 + 2 ln x)

🧮 Calculator

Compute derivative values of eˣ, aˣ, and ln x at a given x.

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🔗 Related Formulas

🎯 Quick Fact

eˣ is the only non-trivial function that is its own derivative. That's why it appears everywhere in physics, finance, and biology — growth processes all follow this rule.