Saturday, October 8, 2016

Unizor - Derivatives - Examples - Exponential Functions





Notes to a video lecture on http://www.unizor.com

Derivative Examples -
Exponential Functions


1. f(x) = ex

f'(x) = ex


Proof
Function increment is
exx − ex =
ex·(eΔx1)
Therefore, to find derivative, we have to find a limit of the following expression as Δx→0:
ex·(eΔx1)/Δx
Since
(eΔx1)/Δx → 1 as Δx → 0,
the derivative is
f'(x) = ex

2. f(x) = ax

f'(x) = ln(a)·ax

Proof
We will use the identity
a = eln(a)
that follows directly from the definition of the natural logarithm.
The function increment looks now as follows:
axx − ax =
ax·(eΔx·ln(a)1)
Therefore, to find derivative, we have to find a limit of the following expression as Δx→0:
ax·(eΔx·ln(a)1)/Δx
Obviously,
(eΔx·ln(a)1)/x·ln(a)) → 1
Therefore,
(eΔx·ln(a)1)/Δx → ln(a)
from which follows that the derivative equals to
f'(x) = ln(a)·ax

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