∫ Math Formulas
29 formulas in 5 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: Derivatives · Integrals · Definite Integral · Limits · Application
d/dx (xⁿ) = n·xⁿ⁻¹
| x | the variable |
|---|---|
| n | the power |
Where it is used: To differentiate power functions — the most-used rule for finding rates of change and slopes.
In simple words: To find how fast xⁿ changes, bring the power down in front and lower the power by one.
Example: d/dx(x³) = 3x²
d/dx (c) = 0
| c | any constant |
|---|
Where it is used: A basic differentiation rule — constants don't change, so their rate of change is zero.
In simple words: A number that never changes has zero rate of change.
Example: d/dx(7) = 0
d/dx(sin x) = cos x, d/dx(cos x) = −sin x
| x | the angle in radians |
|---|
Where it is used: Standard derivatives used in physics (waves, oscillations) and engineering.
In simple words: The rate of change of sine is cosine, and of cosine is negative sine.
Example: slope of sin x at x=0 is cos 0 = 1
d/dx(u·v) = u'v + uv'
| u, v | two functions of x |
|---|---|
| u', v' | their derivatives |
Where it is used: To differentiate the product of two functions — used throughout calculus.
In simple words: For two things multiplied, differentiate the first (keep the second), plus keep the first (differentiate the second).
Example: d/dx(x·sin x) = sin x + x·cos x
d/dx f(g(x)) = f'(g(x)) · g'(x)
| f, g | an outer and inner function |
|---|---|
| f', g' | their derivatives |
Where it is used: To differentiate a function inside another function — essential for composite functions.
In simple words: Differentiate the outside, keep the inside, then multiply by the derivative of the inside.
Example: d/dx (sin(x²)) = cos(x²) · 2x
d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x
| e | Euler's number ≈ 2.718 |
|---|---|
| x | the variable (x>0 for ln) |
Where it is used: Standard derivatives of the exponential and natural-log functions — growth, decay, finance.
In simple words: eˣ is special — its rate of change equals itself; the log function's rate is one-over-x.
Example: slope of ln x at x=2 is 1/2
d/dx(u ± v) = u' ± v', d/dx(k·u) = k·u'
| u, v | functions |
|---|---|
| k | a constant |
Where it is used: To differentiate sums, differences and constant multiples term by term.
In simple words: Differentiate each piece separately; a constant multiplier just tags along.
Example: d/dx(3x² + 5x) = 6x + 5
d/dx(u/v) = (u'v − uv') ÷ v²
| u, v | two functions |
|---|---|
| u', v' | their derivatives |
Where it is used: To differentiate one function divided by another.
In simple words: For a fraction of functions: bottom times derivative of top, minus top times derivative of bottom, all over bottom squared.
Example: d/dx(x/sin x) = (sin x − x cos x)/sin²x
tan→sec²x, cot→−cosec²x, sec→secx tanx, cosec→−cosecx cotx
| x | the angle in radians |
|---|
Where it is used: Standard derivatives of the other trig functions, used across calculus.
In simple words: The rate-of-change formulas for tangent, secant, cotangent and cosecant.
Example: d/dx(tan x) = sec²x
d/dx(√x) = 1/(2√x), d/dx(aˣ) = aˣ ln a
| a | a positive base |
|---|---|
| x | the variable |
Where it is used: Standard derivatives for roots and general exponentials.
In simple words: Handy derivatives for square root and any base raised to x.
Example: d/dx(2ˣ) = 2ˣ ln2
sin⁻¹x → 1/√(1−x²), tan⁻¹x → 1/(1+x²)
| x | the variable |
|---|
Where it is used: Standard derivatives of inverse trig functions (Class 12).
In simple words: The rate-of-change formulas for inverse sine and inverse tangent.
Example: d/dx(tan⁻¹x) = 1/(1+x²)
∫ xⁿ dx = xⁿ⁺¹ ÷ (n+1) + C, n ≠ −1
| x | the variable |
|---|---|
| n | the power |
| C | constant of integration |
Where it is used: To integrate power functions — finding areas, totals from rates, and reversing differentiation.
In simple words: Integration undoes differentiation: raise the power by one and divide by the new power.
Example: ∫ x² dx = x³/3 + C
∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C
| x | the variable |
|---|---|
| C | constant of integration |
Where it is used: Standard integrals used in physics and engineering for areas and totals.
In simple words: The reverse of differentiating: integrating sine gives negative cosine, integrating cosine gives sine.
Example: ∫ cos x dx = sin x + C
∫ (1/x) dx = ln|x| + C, ∫ eˣ dx = eˣ + C
| x | the variable |
|---|---|
| C | constant of integration |
Where it is used: Standard integrals used constantly in calculus, physics and probability.
In simple words: The reverse of the log and exponential derivatives.
Example: ∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ ÷ ln a + C
| a | a positive base ≠ 1 |
|---|---|
| C | constant |
Where it is used: To integrate a general exponential function.
In simple words: The reverse of differentiating aˣ.
Example: ∫ 2ˣ dx = 2ˣ/ln2 + C
∫ sec²x dx = tan x + C, ∫ cosec²x dx = −cot x + C
| x | the variable |
|---|---|
| C | constant |
Where it is used: Standard trig integrals — reverses of the tan and cot derivatives.
In simple words: Integrating sec² gives tan; integrating cosec² gives −cot.
Example: ∫ sec²x dx = tan x + C
∫ dx/√(1−x²) = sin⁻¹x + C, ∫ dx/(1+x²) = tan⁻¹x + C
| x | the variable |
|---|---|
| C | constant |
Where it is used: Standard integrals that produce inverse trig functions.
In simple words: These special fractions integrate to inverse sine and inverse tangent.
Example: ∫ dx/(1+x²) = tan⁻¹x + C
∫ tan x dx = ln|sec x| + C
| x | the variable |
|---|---|
| C | constant |
Where it is used: A standard integral of the tangent function.
In simple words: Integrating tangent gives the log of secant.
Example: used in many area and volume problems
∫ₐᵇ f(x) dx = F(b) − F(a)
| F | an antiderivative of f |
|---|---|
| a, b | the lower and upper limits |
Where it is used: To find the exact area under a curve between two points — physics, probability, economics.
In simple words: Find the antiderivative, then subtract its value at the start from its value at the end.
Example: ∫₀¹ x² dx = 1³/3 − 0 = 1/3
∫ₐᵇ f(x) dx = − ∫ᵇₐ f(x) dx
| a, b | the limits |
|---|
Where it is used: A property of definite integrals used to simplify calculations.
In simple words: Swapping the start and end of an integral just flips its sign.
Example: ∫₀¹ = −∫₁₀
Area = ∫ₐᵇ y dx
| y | the curve's equation |
|---|---|
| a, b | the x-limits |
Where it is used: To find the area enclosed by a curve and the x-axis between two points.
In simple words: Integrating the curve's height across an interval gives the area underneath.
Example: Area under y=x² from 0 to 2 = 8/3
lim (x→0) (sin x ÷ x) = 1
| x | angle in radians approaching 0 |
|---|
Where it is used: A fundamental limit used to derive the derivative of sine and in many proofs.
In simple words: As the angle gets tiny, sine of the angle and the angle itself become practically equal.
Example: For very small x, sin x ≈ x
lim (x→a) (xⁿ − aⁿ)/(x − a) = n·aⁿ⁻¹
| n | the power |
|---|---|
| a | the point |
Where it is used: A fundamental limit used to derive the power rule of differentiation.
In simple words: This limit is exactly the derivative of xⁿ at x = a.
Example: n=2, a=3 → limit = 6
lim(x→0)(eˣ−1)/x = 1, lim(x→0) log(1+x)/x = 1
| x | approaching 0 |
|---|
Where it is used: Standard limits used to differentiate exponential and log functions.
In simple words: For tiny x, eˣ−1 and log(1+x) both behave just like x.
Example: (e^0.001 − 1)/0.001 ≈ 1
lim (x→∞) (1 + 1/x)ˣ = e
| e | ≈ 2.718 |
|---|---|
| x | growing large |
Where it is used: The famous limit that defines Euler's number e — compound interest, growth.
In simple words: Compounding a 100% return over more and more tiny steps approaches e.
Example: (1 + 1/1000)¹⁰⁰⁰ ≈ 2.717
Slope = dy/dx at that point
| dy/dx | the derivative of y with respect to x |
|---|
Where it is used: To find the steepness of a curve at any point — tangents, maxima and minima, velocity.
In simple words: The derivative gives the exact tilt of the curve at whatever point you plug in.
Example: y = x², dy/dx = 2x → slope at x=3 is 6
f'(x) > 0 → increasing; f'(x) < 0 → decreasing
| f'(x) | the derivative |
|---|
Where it is used: To find where a function rises or falls — graphs, optimisation.
In simple words: A positive slope means going up; a negative slope means going down.
Example: f(x)=x²: f'=2x>0 for x>0 → increasing
f'(x) = 0, then f''(x) < 0 → max, f''(x) > 0 → min
| f' | first derivative |
|---|---|
| f'' | second derivative |
Where it is used: To find the highest and lowest points of a function — optimisation problems.
In simple words: Flat slope marks a turning point; the second derivative says if it's a peak or a valley.
Example: f(x)=x²: f'=0 at x=0, f''=2>0 → minimum
Rate = dy/dt (derivative with respect to time)
| y | the changing quantity |
|---|---|
| t | time |
Where it is used: To find how fast something changes with time — speed, growth, filling rates.
In simple words: The derivative with respect to time tells how quickly a quantity is changing.
Example: If s = t², velocity = ds/dt = 2t