Exponential Function — eˣ
The natural exponential function eˣ is the most important function in calculus. Its unique property is that it is its own derivative.
More generally, using the chain rule:
d/dx[e^(3x)] = e^(3x) · 3 = 3e^(3x)
d/dx[e^(x²)] = e^(x²) · 2x = 2x·e^(x²)
d/dx[e^(sin x)] = e^(sin x) · cos x = cos x · e^(sin x)
d/dx[2ˣ] = 2ˣ · ln 2 = 2ˣ ln 2
Logarithmic Function — ln x
The natural logarithm ln x (log base e) has a beautifully simple derivative:
d/dx [ln f(x)] = f'(x) / f(x) (chain rule)
d/dx[ln(x²+1)] = 2x/(x²+1) = 2x/(x²+1)
d/dx[ln(sin x)] = cos x / sin x = cot x
d/dx[log₁₀(x)] = 1/(x · ln 10) = 1/(x ln 10)
Properties of Logarithms (For Differentiation)
These log properties simplify complex expressions before differentiation:
💡 Tip: Before differentiating complex products/quotients/powers — take the natural log first to simplify (logarithmic differentiation, see topic 4).
Summary — All Exp & Log Derivatives
| Function | Derivative | Condition |
|---|---|---|
| eˣ | eˣ | all x |
| e^(f(x)) | f'(x) · e^f(x) | chain rule |
| aˣ | aˣ ln a | a > 0, a ≠ 1 |
| a^(f(x)) | a^f(x) · ln a · f'(x) | chain rule |
| ln x | 1/x | x > 0 |
| ln(f(x)) | f'(x) / f(x) | f(x) > 0 |
| logₐx | 1 / (x ln a) | x > 0 |
Use Product Rule: u = e^(x²), v = ln(3x)
u' = 2x·e^(x²) | v' = 3/(3x) = 1/x
dy/dx = u'v + uv' = 2x·e^(x²)·ln(3x) + e^(x²)·(1/x)
= e^(x²) [2x·ln(3x) + 1/x]