# Fundamental Mathematics/Arithmetic/Differentiation

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## Differentiation

Differentiation is a mathematical operation on an arithmetic function to find the slope of the function's curve .

Differentiation of a function has a mathematical symbol

${\displaystyle {\frac {d}{dt}}f(t)=f^{'}(t)=\lim _{\Delta t\to 0}\sum {\frac {f(t+\Delta t)-f(t)}{\Delta t}}}$

From above,

${\displaystyle df(x)=f^{'}(t)dt=\sum \lim _{\Delta t\to 0}{\frac {f(t+\Delta t)-f(t)}{\Delta t}}dt}$

Meaning differential change of a function is equal to the product of function's gradient multiply by its differential change of function's variable

## Differentiation's Formulas

 Differentiation of a constant ${\displaystyle {\frac {d}{dt}}c=0}$ Differentiation of a constant times time ${\displaystyle {\frac {d}{dt}}at=a}$ Differentiation of a power ${\displaystyle {\frac {d}{dt}}t^{n}=nt^{n-1}}$ Differentiation of a constant times a power ${\displaystyle {\frac {d}{dt}}at^{n}=ant^{n-1}}$

## Example

${\displaystyle {\frac {d}{dt}}4=0}$
${\displaystyle {\frac {d}{dt}}4t=4}$
${\displaystyle {\frac {d}{dt}}t^{n}=nt^{n-1}}$
${\displaystyle {\frac {d}{dt}}4t^{n}=4nt^{n-1}}$