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Contents
Exponential Function
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Exponential Function  Detailed
treatment of the manipulation of exponential and logarithmic functions can be
found in our relevant section. In this section, knowledge of this is expected.
Derivative
of the Exponential Function
The
exponential function, ex, is unique, as its derivative is itself.
Given y = ex , then
In general, for y = ef(x) ,
Given
y = ax, then
In general, for y = af(x),
Proving
the Derivatives
The
first principles method can be used to prove the derivative of ex.
However, the limit is hard to evaluate. Given the limit, the proof is easy.
Let
y = f(x) = ex
This
result, in turn, can be used to prove the derivative of ax.
d/dx (ax) = d/dx (ex ln a)
properties
of logarithms
= ex ln a. d/dx (x ln a)
= exln a. ln a
= ax. ln a
Examples
 Differentiate
with respect to x:
(i) e3x
(ii) e2x + e-x
(iii) e2x (x - 1)
(iv) 35x+4
(x)
Solutions:
(i)
d/dx (e3x) = 3e3x
(ii)
d/dx (e2x + e-x) = 2e2x - e-x
(iii)
d/dx [e2x (x - 1)] = 2e2x(x - 1) + e2x
= e2x [2(x - 1) + 1]
= e2x(2x + 1)
(iv)
d/dx (35x+4) = 5.ln3.35x+4
(x)
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