Leibniz's notation for differentiation

The derivative of the function f(x) is written:

<math>\frac{d\left(f\left(x\right)\right)}{dx}</math>

If we have a variable representing a function, for example if we set:

<math>y = f\left(x\right)</math>

then we can write the derivative as:

<math>\frac{dy}{dx}</math>

Using Newton's notation for differentiation, we can write:

<math>\frac{d\left(f\left(x\right)\right)}{dx} = f'\left(x\right)</math>

Using the dot notation for differentiation, we can write:

<math>\frac{dx}{dt} = \dot{x}</math>

For higher derivatives, we express them as follows:

<math>\frac{d^n\left(f\left(x\right)\right)}{dx^n}</math> or <math>\frac{d^ny}{dx^n}</math>

denotes the nth derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is:

<math>\frac{d \left(\frac{d \left( \frac{d \left(f\left(x\right)\right)} {dx}\right)} {dx}\right)} {dx}</math>

which we can loosely write as:

<math>\left(\frac{d}{dx}\right)^3 \left(f\left(x\right)\right) =

Now drop the brackets and we have:

<math>\frac{d^3}{dx^3}\left(f\left(x\right)\right)</math> or <math>\frac{d^3y}{dx^3}</math>

The chain rule and integration by subsitution[?] rules are especially easy to express here, because the "d" terms appear to cancel:

<math>\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dw} \cdot \frac{dw}{dx}</math> etc.

and:

<math>\int y dx = \int y \frac{dx}{du} du</math>