Differentiation, integration and complex numbers — watch each relationship animate live, with sliders so you can test it yourself.
All Rules
Power Rule
Trig
Chain Rule
Integration
Complex Numbers
Reference tables for the main differentiation rules and standard derivatives. Click the tabs above for an animated visualisation of the Power Rule, Trig derivatives, and Chain Rule.
Differentiation Rules
Riail an toraidh
y = uv
⇒dydx = udvdx + vdudx
Product rule
Riail an lín
y = uv
⇒dydx = vdu/dx − udv/dxv²
Quotient rule
Cuingriail
f(x) = u(v(x))
⇒ f'(x) = dudvdvdx
Chain rule
Standard Derivatives
f(x)
f'(x)
xn
nxn−1
ln x
1/x
ex
ex
eax
aeax
ax
ax ln a
cos x
−sin x
sin x
cos x
tan x
sec²x
cos−1(x/a)
−1/√(a²−x²)
sin−1(x/a)
1/√(a²−x²)
tan−1(x/a)
a/(a²+x²)
Standard Integrals
f(x)
∫ f(x) dx
xn (n ≠ −1)
xn+1/(n+1) + c
1/x
ln|x| + c
ex
ex + c
eax
(1/a)eax + c
ax
ax/ln a + c
cos x
sin x + c
sin x
−cos x + c
tan x
ln|sec x| + c
cos²x
x/2 + (sin 2x)/4 + c
sin²x
x/2 − (sin 2x)/4 + c
1/√(a²−x²)
sin−1(x/a) + c
1/(a²+x²)
(1/a)tan−1(x/a) + c
1/(a²−x²)
(1/2a)ln|(a+x)/(a−x)| + c
1/(x²−a²)
(1/2a)ln|(x−a)/(x+a)| + c
Integration by parts: ∫u dv = uv − ∫v du (from the official Leaving Cert Formulae & Tables booklet)
Power Rule
d/dx [x²] = 2x
The gold tangent on f(x) has a slope equal to the height of the red f'(x) at the same x. Change n to see how the derivative changes.
f(x) = x²
f'(x) = 2x
n = 2
x = —
slope = —
f(x) =
Trig Derivatives
d/dx [sin x] = cos x
The slope of sin x at every x equals the value of cos x there. When sin x peaks, its slope is zero — and cos x = 0 at the same point.
f(x) = sin x
f'(x) = cos x
x = —
slope = —
f(x) =
Chain Rule
d/dx[f(g(x))] = f'(g(x))·g'(x)
Derivative of the outer function (keeping inner intact) × derivative of the inner function. Below: y = sin(x²) as a live example.
y = sin(x²) | Outer: f(u) = sin u → f'(u) = cos u | Inner: g(x) = x² → g'(x) = 2x ∴ dy/dx = cos(x²) · 2x
f(x) = sin(x²)
f'(x) = 2x·cos(x²)
x = —
slope = —
f(x) =
Integration
∫x² dx = x³/3 + c
As t sweeps across, the shaded area under f(x) from 0 to t grows. Its value at every t equals the height of F(t) on the right — the Fundamental Theorem of Calculus in action.
f(x) = x²
F(t) = t³/3
n = 2
t = —
area = —
f(x) =
Demo: Sine & Circle Area
Sine & Circle Area
∫₀^θ sin(x) dx = 1 − cos θ
As P sweeps round the unit circle, the pie-slice it sweeps grows steadily and fills the whole circle (area = πr²) by θ = 2π. But the signed area under sin(x) on the right rises then an equal negative hump cancels it out — so ∫₀^2π sin(x) dx = 0. The circle's area comes from the radius sweeping round, not from the height of sin(x).
Unit circle — sector swept by radius
sin(x), shaded 0 → 2π
θ = —
sector area = —
∫₀^θ sin(x)dx = —
circle area = π ≈ 3.142
Complex Numbers
(cos θ + i sin θ)² = cos 2θ + i sin 2θ
De Moivre's Theorem: raising z = cos θ + i sin θ to the power n multiplies its argument by n. Watch the point on the right spin n times faster around the unit circle.