Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 13/Differentials

Differentials

Let $y = \map f x$ and $\Delta y = \map f {x + \Delta x} - \map f x$. Then:

$13.49$: Definition of Differential: $\dfrac {\Delta y} {\Delta x} = \dfrac {\map f {x + \Delta x} - \map f x} {\Delta x} = \map {f'} x + \epsilon = \dfrac {\d y} {\d x} + \epsilon$

where $\epsilon \to 0$ as $\Delta x \to 0$. Thus:

$13.50$: $\Delta y = \map {f'} x \Delta x + \epsilon \Delta x$

If we call $\Delta x = \d x$ the differential of $x$, then we define the differential of $y$ to be:

$13.51$: Definition of Differential of $y$: $\Delta y = \map {f'} x \Delta x + \epsilon \Delta x$