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Elementary Differential and Integral Calculus FORMULA SHEET
Exponents xa^ · xb^ = xa+b, ax^ · bx^ = (ab)x, (xa)b^ = xab, x^0 = 1.
Logarithms ln xy = ln x + ln y, ln xa^ = a ln x, ln 1 = 0, eln^ x^ = x, ln ey^ = y, ax^ = ex^ ln^ a.
Trigonometry cos 0 = sin π 2 = 1, sin 0 = cos π 2 = 0 , cos^2 θ + sin^2 θ = 1, cos(−θ) = cos θ, sin(−θ) = − sin θ, cos(A + B) = cos A cos B − sin A sin B, cos 2θ = cos^2 θ − sin^2 θ, sin(A + B) = sin A cos B + cos A sin B, sin 2θ = 2 sin θ cos θ,
tan θ = sin^ θ cos θ
, sec θ = 1 cos θ
, 1 + tan^2 θ = sec^2 θ.
Inverse Functions y = sin−^1 x means x = sin y and −π 2 6 y 6 π 2. y = cos−^1 x means x = cos y and 0 6 y 6 π. y = tan−^1 x means x = tan y and −π 2 < y < π 2. y = x^1 /n^ means x = yn. y = ln x means x = ey.
Alternative Notation arcsin x = sin−^1 x, arccos x = cos−^1 x, arctan x = tan−^1 x, loge x = ln x. Note: sin−^1 x 6 = (sin x)−^1 , cos−^1 x 6 = (cos x)−^1 , tan−^1 x 6 = (tan x)−^1. However: sin^2 x = (sin x)^2 , cos^2 x = (cos x)^2 , tan^2 x = (tan x)^2.
Lines The line y = mx + c has slope m. The line through (x 1 , y 1 ) with slope m has equation y − y 1 = m(x − x 1 ).
The line through (x 1 , y 1 ) and (x 2 , y 2 ) has slope m = y^2 −^ y^1 x 2 − x 1
and equation y^ −^ y^1 x − x 1
= y^2 −^ y^1 x 2 − x 1
The line y = mx + c is perpendicular to the line y = m′x + c′^ if mm′^ = − 1.
Circles The distance between (x 1 , y 1 ) and (x 2 , y 2 ) is
(x 1 − x 2 )^2 + (y 1 − y 2 )^2. The circle with centre (a, b) and radius r is given by (x − a)^2 + (y − b)^2 = r^2.
Triangles In a triangle ABC:
(Sine Rule) a sin A
= b sin B
= c sin C
; (Cosine Rule) a^2 = b^2 + c^2 − 2 bc cos A. 4
Pascal’s Triangle (x + y)^2 = x^2 + 2 xy + y^2 , (x + y)^3 = x^3 + 3 x^2 y + 3 xy^2 + y^3 and so on. The coefficients in (x + y)n^ form the nth row of Pascal’s triangle:
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
............. and so on.
Quadratics
If ax^2 + bx + c = 0 , with a 6 = 0 , then x = −b^ ±
b^2 − 4 ac 2 a.
Calculus If y = u + v then dy dx
= du dx
. If y = uv then dy dx
= du dx
v + u dv dx
If y =
u v then^
dy dx =
{du dx v^ −^ u
dv dx
v^2. ∫ (u + v) dx =
u dx +
v dx.
u dx dvdx = uv −
∫ (^) du dx v dx. If y is a function of u where u is a function of x, then dy dx =^
dy du
du dx and
ydudx dx =
y du.
Standard Derivatives and Integrals
If y = xa^ then dydx = a xa−^1 ; and
xa^ dx = x
a+ 1 a + 1 +^ constant^ (a^6 = −^1 ).
If y = sin x then dydx = cos x; and
sin x dx = − cos x + constant.
If y = cos x then dydx = − sin x; and
cos x dx = sin x + constant.
If y = tan x then
dy dx =^ sec
(^2) x; and
tan x dx = ln | sec x| + constant.
If y = ex^ then dydx = ex; and
ex^ dx = ex^ + constant.
If y = ln x then dydx = (^) x^1 ; and
x dx^ =^ ln^ |x|^ +^ constant.
If y = sin−^1 x then dy dx
= √^1
1 − x^2
; and
1 − x^2
dx = sin−^1 x + constant.
If y = cos−^1 x then dydx = √−^1 1 − x^2
If y = tan−^1 x then dy dx
1 + x^2
; and
1 + x^2
dx = tan−^1 x + constant. 5
