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Problems for "The Number e and the Natural Log"
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► Problems for "Derivatives of ex and of the Natural Log"
Inverse, Exponential, and Logarithmic Functions
Derivatives of ex and of the Natural Log
Derivatives of Exponential Functions
One truly remarkable characteristic of ex is that
ex = ex
Besides the trivial case of f (x) = 0, ex and its constant multiples are the only functions whose derivatives are equal
to themselves!
Incorporating the principles of the chain rule, we might also say that if u is a function
of x, then
eu = eu
By extension, exdx = ex + c. Using the fact that
eu = eu
we can derive a more general formula for the derivative of ax, where a is any
positive constant.
First, note that ax can be rewritten as
eln(ax)
since ex and ln(x) are inverses of each other.
This quantity can also be written as
ex ln(a)
.
Now,
ax = ex ln(a)
By the chain rule,
ax = ex ln(a)ln(a)
Changing the exponential function back to its original form, we get
ax = axln(a)
By extension,
axdx=ax+c
Derivatives of Logarithms
It may be satisfying to learn now that for x>0,
ln(x) =
The appeal rests in the corresponding implication that
=lnx+c
Recall that the power rule did not offer a way of integrating the function ,
but now it is possible to do so.
A related rule for logarithms of any base is that
loga(x) =
Logarithmic Differentiation
To find the derivative of a constant raised to a power of x, the rule presented earlier in
this section should suffice. However, to find the derivative of a function of x that is
raised to a power of x, the technique of logarithmic differentiation is necessary.
Example: Differentiate y = x3x.
Step One: Take the natural log of both sides of the equation:
ln(y) = ln(x3x).
Step Two: Now use log rules to take the variable x out of the exponent and
turn it into a product: ln(y) = (3x)(ln(x)).
Step Three: Implicitly differentiate both sides with respect to x (remember to use the
chain rule):
ex
= ex
exdx = ex + c
ax
= 
ex ln(a)
ln(a)
axdx=
ax+c
x
+c
= 3x
+3 ln(x)
3+3 ln(x)
y
x3x
