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

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Let $y = f \left({x}\right)$ and $\Delta y = f \left({x + \Delta x}\right) - f \left({x}\right).$ Then:

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

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

$13.50$: $\Delta y = f' \left({x}\right) \Delta x + \epsilon \Delta x$

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 $y$: $\Delta y = f' \left({x}\right) \Delta x + \epsilon \Delta x$