Core Derivative Formulas: Quick Reference for GCSE, A-Level and Calculus

This page collects some of the most important derivative formulas. All functions are assumed to be differentiable where the formulas are applied.

1. Basic Derivatives

Function Derivative
d/dx (c) 0
d/dx (x) 1
d/dx (kx) k
d/dx (xn) n xn-1  (for any real constant n)
d/dx (x2) 2x
d/dx (x3) 3x2
d/dx (1/x) -1/x2
d/dx (√x) 1/(2√x)

2. Exponential and Logarithmic Functions

Function Derivative
d/dx (ex) ex
d/dx (ax) ax ln(a)
d/dx (ln|x|) 1/x  (x ≠ 0)
d/dx (loga|x|) 1 / (x ln(a))  (x ≠ 0, a > 0, a ≠ 1)

3. Trigonometric Functions

Function Derivative
d/dx (sin x) cos x
d/dx (cos x) -sin x
d/dx (tan x) sec2 x
d/dx (cot x) -cosec2 x
d/dx (sec x) sec x tan x
d/dx (cosec x) -cosec x cot x

4. Inverse Trigonometric Functions

Function Derivative
d/dx (sin-1 x) 1 / √(1 - x2)
d/dx (cos-1 x) -1 / √(1 - x2)
d/dx (tan-1 x) 1 / (1 + x2)

5. Hyperbolic Functions

Function Derivative
d/dx (sinh x) cosh x
d/dx (cosh x) sinh x
d/dx (tanh x) sech2 x

6. General Differentiation Rules

Rule Formula
Constant multiple d/dx (k f(x)) = k f′(x)
Sum / difference d/dx (f(x) ± g(x)) = f′(x) ± g′(x)
Product rule d/dx (f(x) g(x)) = f′(x) g(x) + f(x) g′(x)
Quotient rule d/dx (f(x)/g(x)) = (f′(x) g(x) - f(x) g′(x)) / (g(x))2
Chain rule d/dx (f(g(x))) = f′(g(x)) · g′(x)

These formulas cover the core derivatives used in most GCSE, A-Level and early university calculus work.

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