Derivatives of common functions
Once you know the derivatives of the basic functions and the rules (product, quotient, chain), you can differentiate almost anything.
Polynomial functions
Use the power rule on each term of a polynomial function.
| f(x) | f′(x) |
|---|---|
| 3x⁵ − 2x² + x − 9 | 15x⁴ − 4x + 1 |
| (x² + 1)² = x⁴ + 2x² + 1 | 4x³ + 4x |
Exponential functions
The natural exponential eˣ (e ≈ 2.718) is special: it’s its own derivative.
| f(x) | f′(x) |
|---|---|
| eˣ | eˣ |
| e^(kx) | k·e^(kx) |
| e^(g(x)) | g′(x)·e^(g(x)) (chain rule) |
| bˣ | bˣ ln b |
Examples:
- d/dx e^(3x) = 3e^(3x)
- d/dx 2ˣ = 2ˣ ln 2
- d/dx e^(x²) = 2x e^(x²)
The slope of y = eˣ at any point equals its height. That’s why e appears in growth and decay models.
Logarithmic functions
| f(x) | f′(x) |
|---|---|
| ln x (the natural logarithm) | 1/x |
| ln(g(x)) | g′(x)/g(x) |
| log_b x | 1/(x ln b) |
Example: d/dx ln(x² + 1) = 2x/(x² + 1)
Trigonometric functions
(x in radians)
| f(x) | f′(x) |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
Examples:
- d/dx sin(3x) = 3 cos(3x)
- d/dx cos(x²) = −2x sin(x²)
- d/dx x sin x = sin x + x cos x (product rule)
Why radians?
These formulas rely on lim (x→0) sin x/x = 1, which is only true in radians. In degrees, extra factors of π/180 would appear.
Combined examples
Example: y = e^(2x) cos x
y′ = 2e^(2x) cos x − e^(2x) sin x = e^(2x)(2 cos x − sin x)
Example: y = ln(sin x)
y′ = cos x / sin x = cot x
Applications
- Exponential decay: N = N₀e^(−kt) → dN/dt = −kN (rate proportional to amount)
- Simple harmonic motion: x = A sin(ωt) → velocity = Aω cos(ωt)
Worked examples
Example 1: d/dx (4eˣ − 3x²) = 4eˣ − 6x.
Example 2: d/dx 5 sin x = 5 cos x.
Example 3: d/dx ln(3x) = 3/(3x) = 1/x.
Common mistakes
- Using the power rule on eˣ (it’s not xⁿ).
- Sign error with cos: d/dx cos x = −sin x (note the minus sign).
- Forgetting the chain rule factor for e^(kx) or sin(kx).
Try it yourself
- d/dx (x⁴ − 2eˣ)
- d/dx e^(−5x)
- d/dx ln(x³)
- d/dx cos(4x)
- d/dx x² eˣ
Answers: 1) 4x³ − 2eˣ 2) −5e^(−5x) 3) 3/x 4) −4 sin(4x) 5) eˣ(x² + 2x)
Quick recap
- Polynomial functions: power rule term by term.
- Natural exponential: d/dx eˣ = eˣ; natural logarithm: d/dx ln x = 1/x.
- Trigonometric functions (in radians): sin → cos, cos → −sin, tan → sec².