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delitzo1d4

Answered question

2022-04-12

How do you use the definition of a derivative to find the derivative of

f (x) = e^{x}

Answer & Explanation

o2z1zsjj0

Beginner2022-04-13Added 10 answers

Using

\frac{d f}{d x} = lim_{△ x \to 0} \frac{f (x + △ x) - f (x)}{△ x}

Explanation:

\frac{d}{d x} e^{x} = lim_{△ x \to 0} \frac{e^{x + △ x} - e^{x}}{△ x} = lim_{△ x \to 0} e^{x} (\frac{e^{△ x} - 1}{△ x})

△ x \to 0, e^{△ x} \to 1^{+}

The ratio approximates to 1 hence
The derivative of

e^{x}

e^{x}

Jax Burns

Beginner2022-04-14Added 13 answers

$f (x) = e^{x}$
$lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = \frac{e^{x + h} - e^{x}}{h}$
from power rules
$= lim_{h \to 0} \frac{e^{x} e^{h} - e^{x}}{h}$
you can use taylor series to have an excellent approximation of $e^{h}$ when $h \to 0$ which is
$e^{h} = h + 1$ when $h \to 0$
$= lim_{h \to 0} \frac{e^{x} (h + 1) - e^{x}}{h} = lim_{h \to 0} \frac{h e^{x} + e^{x} - e^{x}}{h}$
which is $lim_{h \to 0} e^{x} = e^{x}$

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