Derivative & Multiplication Symbols

Here we have a key to help decipher what lots of our mathematical symbols mean. Across the world, mathematical symbols can be written slightly differently even though they represent the same thing. The purpose of this key is to clear up these differences to allow everyone to learn despite having been taught slight differences in the use of symbols.

This may also help remind us what some symbols represent if we have forgotten them or if our teacher has assumed we know what they mean when we actually don't.

Multiplication

There are many ways of symbolizing multiplication and it can even be assumed that, in certain situations, numbers and variables written beside each other, without a symbol, can mean multiplication.

Common Symbols For Multiplication

x

one of the first symbols we are introduced to, it is often not used to avoid confusing it with x when it is being used as a variable. If we were to multiply 2 by 4 it would look like 2 x 4

⋅   or   hollow circle symbolizing multiplication

a small hollow or solid "floating" dot, not to be confused with a decimal point or a full-stop. If we were to multiply 2 by 4 it would look like 2 ⋅ 4

( ) [ ]

When two or more numbers or letter-symbols are written beside each other within brackets, it is assumed that it means multiplication. If we were to multiply 2 by 4 it would look like (2)(4). Brackets may also be used to emphasize something or to help us distinguish negative numbers. For example, if we had the equation y = -2⋅x + 1 and we subbed in x = -3, brackets would help distinguish the negative values. e.g. y = -2(-3) + 1

written beside each other without a symbol

Often we will see expressions written like 2x or 5a or 2gh. When we have a number written beside a letter with no symbols, it is assumed that this means multiplication. You may also see just letters written beside each other. For example, Force being equal to mass time acceleration being written as f = ma

Mathematicians are lazy and like to remove any extra work that is deemed unnecessary!

To review, the table below equates different ways of writing multiplication.

words x ⋅   or   hollow circle symbolizing multiplication brackets no symbols
two times four 2 x 4 2⋅4 (2)(4) 24 *
three times eight 3 x 8 3⋅8 (3)(8) 38 *
one half times 5 ½ x 5 ½ ⋅ 5 (½)(5) ½5 *
six times "X" 6 x X 6⋅X (6)(X) 6X
two times one times five 2 x 1 x 5 2⋅1⋅5 (2)(1)(5) 215 *
seven times "a" 7 x a 7⋅a (7)(a) 7a
mass (m) times acceleration (a) m x a m⋅a (m)(a) ma
"g" times "h" times "p" g x h x p g⋅h⋅p (g)(h)(p) ghp

* - does not work!

Differentiation

Like multiplication, differentiation has different symbols for showing the derivative. below are some you will see alongside why they may be used.

dy over dx derivative operator

At its purest form, this means the change in y over the change in x, or dy by dx, which is what the derivative means. It is used when we have the derivative of, say, y = "something with x." If we had something like, a = "something with b" e.g. a = 2b + 1, then the symbol for the derivative of a would be da over db derivative operator

f'(x)

used to symbolize the derivative of a function of x. The derivative of (f(x) would be symbolized by f'(x). It is different from dy/dx as shown above (which involves a "y". A function of x is purely to do with one variable, x, and f'(x) is the way we symbolize it. For example, if we had y = 2x + 1 and wanted to write its derivative, it would be incorrect to write, f'(x) = (1)2(x0) and f'(x) = 2

The correct symbolization would be: dy over dx derivative operator = 2

D(f(x))

is another way of writing f'(x). if we had f(x) = 3x2 + 1 we could also write the derivative of f(x) as, D(f(x)) or D(3x2 + 1)

This method is often used for the derivative of an expression. e.g. D(expression). It is a simple way of writing a derivative.