Leibniz's notation for differentiation
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See Leibniz notation and separation of variables for, among other things, an account of certain advantages of this notation over others.
In 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 Lagrange's notation for differentiation, we can write:
- <math>\frac{d\left(f\left(x\right)\right)}{dx} = f'\left(x\right)<math>
Using the Newton's 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) =
\frac{d^3}{\left(dx\right)^3} \left(f\left(x\right)\right)<math>
Now drop the brackets and we have:
- <math>\frac{d^3}{dx^3}\left(f\left(x\right)\right)\ \mbox{or}\ \frac{d^3y}{dx^3}<math>
The chain rule and integration by substitution 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>