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O-Level Additional Mathematics Calculus and Algebra Rules

Differentiation and integration results, plus the main algebra, logarithm and trigonometry rules for O-Level Additional Mathematics.

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Derivative of x^n
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1
Derivative of x^n
n x^(n - 1).
2
Derivative of e^(kx)
k e^(kx).
3
Derivative of ln x
1/x.
4
Derivative of sin x
cos x.
5
Derivative of cos x
-sin x.
6
Derivative of tan x
sec^2 x.
7
Product rule
d(uv)/dx = u dv/dx + v du/dx.
8
Quotient rule
d(u/v)/dx = (v du/dx - u dv/dx) / v^2.
9
Chain rule
dy/dx = dy/du x du/dx.
10
Stationary point
A point where dy/dx = 0.
11
Nature of a stationary point
If d2y/dx2 is positive it is a minimum, if negative a maximum.
12
Equation of a tangent
y - y1 = m (x - x1), where m = dy/dx at the point.
13
Gradient of the normal
-1 divided by the gradient of the tangent.
14
Integral of x^n
x^(n + 1)/(n + 1) + c, for n not equal to -1.
15
Integral of 1/x
ln|x| + c.
16
Integral of e^(kx)
e^(kx)/k + c.
17
Integral of cos kx
(1/k) sin kx + c.
18
Integral of sin kx
-(1/k) cos kx + c.
19
Area under a curve
The definite integral of y dx between the limits, taken above the x-axis.
20
Velocity and acceleration
v = ds/dt and a = dv/dt. Integrate to go back.
21
Remainder theorem
The remainder when P(x) is divided by x - a is P(a).
22
Factor theorem
x - a is a factor of P(x) if and only if P(a) = 0.
23
Discriminant
b^2 - 4ac. Positive: two real roots. Zero: equal roots. Negative: no real roots.
24
Sum and product of roots
Sum = -b/a and product = c/a.
25
Completing the square
x^2 + bx + c = (x + b/2)^2 + c - b^2/4.
26
Logarithm laws
log xy = log x + log y; log (x/y) = log x - log y; log x^n = n log x.
27
Sine rule and cosine rule
a/sin A = b/sin B; c^2 = a^2 + b^2 - 2ab cos C.
28
sin^2 x + cos^2 x
1.
29
Arc length and sector area
s = r theta and A = 1/2 r^2 theta, with theta in radians.
30
Binomial theorem
The coefficient of x^r in (1 + x)^n is nCr.

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