The derivative of a function is another function which represents the slope or gradient of that function. It can also be used to represent the rate of change of one variable with respect to another - for example acceleration, the derivative of velocity, is the rate of change of velocity with respect to time.

There are a number of ways of representing the derivative of an equation. Usually these depend on how you represented the function in the first place. \(\frac{d}{dx}\) is the way of representing the differentiation operative i.e. if you see \( \frac{d}{dx} ( \text{Some function)}\) then you should differentiate the function you see in the brackets.

The following list shows the function on the left, and the corresponding way of writing its derivative on the right.

\[\frac{d}{dx} f(x) = f'(x)\]

\[\frac{d}{dx}( y )= \frac{dy}{dx} \text { or y' } \]

If you are differentiating with respect to a variable other than \( x\) ,let's say \( t\) , do not use\( \frac{d}{dx}\) but \( \frac{d}{dt}\) .

Introduction to Differentiation