Proof Archive

Adding exponents

This proofs shows that you can add exponents when multiplying powers with the same base.

Chain rule in calculus

This proofs show that the derivative of a nested function is the derivative of the outer function multiplied by the derivative of the inner function.

Coefficient rule in calculus

This proof shows that the derivative of a function with a coefficient is the coefficient times the derivative of that function.

Constant rule in calculus

This proofs shows that the derivative of a constant is always zero.

Exponent rule in calculus

This proofs show the derivative of a^x is a^x * ln(a).

Log rule in calculus

This proofs shows the derivative of a logarithmic function.

Power rule in calculus

This proofs shows that the derivative of x^r is r * x^(r - 1).

Product rule in calculus

This proof shows that the derivative of a product g*j is g'*j + g*j'.

Quotient rule in calculus

This proof shows that the derivative for the quotient or fraction a/b is (a'b - ab') / b^2.

Sum rule in calculus

This proofs shows that the derivative of two functions is the derivative of the first function plus the derivative of the second function.

Change of base formula for logarithms

This proofs shows that every logarithm can be written as a division of two logarithms with a custom base.

Cosine rule

This proofs shows that the cosine rule is correct.

The cube root of a complex number

The definition of the derivative

This proofs shows what the definition of the derivative is and how it came about

Derivative of cos(x)

This proof shows what the derivative of cos(x) is.

The derivative of e to the x

This proof shows that the definition of e^x is e^x.

The derivative of ln(x)

This proof shows that the derivative of ln(x) is 1/x.

Derivative of sin(x)

This proof shows what the derivative of sin(x) is.

Derivative of tan(x)

This proof shows what the derivative of tan(x) is.

The difference of logarithms

This proof shows that the difference of two logarithms with the same base is just one logarithm with the inside parts being divided.

Distance formula between a point and a line

This proof shows a formula for calculating the distance between a point and a line in the form of y = ax + b.

Writing an even number as an integer

This proofs shows how to write an even number as an integer.

An even number plus an even number is even

This proofs shows that an even number plus another even number will always be even.

An even number plus an odd number is odd

This proofs shows that an even number plus an odd number will always be odd.

An even number times an even number is even

This proofs shows that an even number times an even number will always be even.

An even number times an odd number is even

This proofs shows that an even number times an odd number will always be even.

Exact values of the trig functions

This table shows all exact values of sine, cosine and tangent.

Factoring the difference of two cubes

This proof shows how to factor a difference of two cubes.

Factoring the difference of two squares

This proof shows how to factor a difference of two squares.

Factoring the sum of two cubes

This proof shows how to factor a sum of two cubes.

Heron's formula: area of triangle

This shows Heron's formula for finding the area of any triangle.

There are infinitely many natural numbers

This proof shows that there are infinitely many natural numbers.

The keep-change-flip rule for fractions

This proof shows the famous keep-change-flip rule when dividing by fractions.

The limit of sin(x) over x

This proof shows that the limit as x goes to zero of sin(x) over x converges to one.

The logarithm power rule

This proofs shows that a logarithm of a power is the same as the exponent multiplied by that logarithm.

Multiplying exponents

This proofs shows that the exponents can be multiplied when a power is raised to another power.

Writing an odd number as an integer

This proofs shows how to write an odd number as an integer.

An odd number plus an odd number is even

This proofs shows that an odd number plus an odd number will always be even.

An odd number times an odd number is odd

This proofs shows that an odd number times an odd number will always be odd.

Pythagorean theorem

This proofs shows how the famous Pythagorean theorem works: a^2 + b^2 = c^2

Quadratic formula

This proof shows how the famous quadratic formula works.

Angle sum/difference formula for sine and cosine

This proof shows the formula for adding and subtracting angles for sine and cosine.

Sine rule

This proof shows that the ratio between a triangle's side length and the sine of its angle is the same for all sides.

Sine squared plus cosine squared is always one

This proof shows that sine squared plus cosine squared is always one, using the Pythagorean theorem.

The square root of a prime number is always irrational

This proofs shows that the square root of a prime number is always irrational.

The sum of logarithms

This proof shows that the sum of two logarithms with the same base is just one logarithm with the inside parts being multiplied.

Tangent is sine divided by cosine

This proof shows that sine divided by cosine is tangent.

Trig values of 15 degrees

This proof shows the exact value of sin(15), cos(15) and tan(15)

Trig values of 18 degrees

This proof shows the exact value of sin(18), cos(18) and tan(18)

Trig values of 3 degrees

This proof shows the exact value of sin(3), cos(3) and tan(3).

Trig values of 30 and 60 degrees

This proof shows the exact value of sin(30), cos(30), tan(30), sin(60), cos(60) and tan(60).

Trig values of 36 degrees

This proof shows the exact value of sin(36), cos(36) and tan(36)

Trig values of 45 degrees

This proof shows the exact value of sin(45), cos(45) and tan(45)

Trig values of 54 degrees

This proof shows the exact value of sin(54), cos(54) and tan(54)