Unizor - Derivatives - Examples - Exponential Functions

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

Derivative Examples -
Exponential Functions

1. f(x) = e^x

f'(x) = e^x


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

2. f(x) = a^x

f'(x) = ln(a)·a^x

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