A
notation[?] for describing
derivatives.
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) =
\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)</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>
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