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Table Of Derivatives

A reference table of derivative rules and common derivatives (including trig, exponential, logarithmic, inverse trig, and hyperbolic functions).

  • GCE A-Level H2 Physics 2027
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This page is a reference table. In most physics problems, mistakes come from missing a chain-rule factor or mixing up a sign.

Rules on differentiation

  • Product rule: (d/dx)(uv) = vdu/dx + udv/dx
  • Quotient rule: (d/dx)(u/v) = (vdu/dx-udv/dx)/v²
  • Chain rule: d/dx[f(u)] = (df/du)du/dx

For reference (integration):

  • Integration by parts: ∫ u dv = uv-∫ v du
Two fast habits that prevent most errors
  • If you see a function inside another function, write the inner derivative explicitly (chain rule).
  • For logs: d/dx ln(f) = f'/f.

Quick worked examples

A. (d/dx)(x² e^3x)

Use product rule + chain rule:

(d/dx)(x² e^3x) = 2x e^3x + x² · 3e^3x = e^3x(2x + 3x²).

B. (d/dx)[ln(sin x)]

Use d/dx ln(f) = f'/f:

d/dx ln(sin x) = (cos x)/(sin x) = cot x.

Basic Properties of Derivatives

(d/dx)(cf(x)) = cf'(x)

d/dx (xⁿ) = nxⁿ⁻¹

(fg)' = f'g + fg'

d/dx (f(g(x))) = f'(g(x))g'(x)

d/dx (ln g(x)) = g'(x)/g(x)

(f(x) ± g(x))' = f'(x) ± g'(x)

(d/dx)(c) = 0

(f/g)' = (f'g - fg')/g²

d/dx (e^(g(x))) = g'(x) e^(g(x))

Note: c is any constant, n is any number

Standard Derivatives:

Polynomials

d/dx (c) = 0

(d/dx)(x) = 1

d/dx (cx) = c

d/dx (xⁿ) = nxⁿ⁻¹

d/dx (cxⁿ) = ncxⁿ⁻¹

Trig. Functions

d/dx (sin x) = cos x

d/dx (cos x) = - sin x

d/dx (tan x) = sec² x

d/dx (sec x) = sec x tan x

d/dx (csc x) = - csc x cot x

d/dx (cot x) = - csc² x

Inverse Trig. Functions

d/dx (sin⁻¹ x) = 1/(square root of (1 - x²))

d/dx (cos⁻¹ x) = - 1/(square root of (1 - x²))

d/dx (tan⁻¹ x) = 1/(1 + x²)

d/dx (sec⁻¹ x) = 1/(‖x‖ square root of (x²-1))

d/dx (csc⁻¹ x) = - 1/(‖x‖ square root of (x²-1))

d/dx (cot⁻¹ x) = - 1/(1 + x²)

Exponential/Logarithm Functions

d/dx (a^x) = a^x ln(a)

d/dx (e^x) = e^x

d/dx (ln(x)) = 1/x, x > 0

d/dx (ln ‖x‖) = 1/x, x ≠ 0

d/dx (logₐ (x)) = 1/(x ln a), x > 0

Hyperbolic Trig. Functions

d/dx (sinh x) = cosh x

d/dx (cosh x) = sinh x

d/dx (tanh x) = sech² x

d/dx (sech x) = - sech x tanh x

d/dx (csch x) = - csch x coth x

d/dx (coth x) = - csch² x

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