MTH 226 Derivatives of Logarithms Calculus Worksheet

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1. Derivatives of logarithms
y = ex
y = ln x
1
y = x1
1
Recall that the function x 7→ ln x is the inverse function of the Exponential function. If
particular, if ln x = y then ey = x. Using the Implicit Differentiation techniques from the
previous lecture we obtain
Theorem 1.
d
(ln x) = x1 .
dx
which is valid for all x > 0.
To prove the theorem, we start by differentiating the equation
ey = x
dy
with respect to x (where y = ln x) and then solving it for dx
.
d
(ey ) = 1.
dx
dy
ey · dx
= 1,
dy
=
dx
1
= x1 .
ey
2
2. General Logarithms.
Let a be a positive constant. Recall that x 7→ loga x is the inverse of x 7→ ax . Since general
logarithms can be easily converted to natural logs by
x
loga x = ln
,
ln a
d
d
we may derive the derivative of the general log. dx
(loga x) = dx
( ln x )
ln a
d
1
= ln1a · dx
ln x = x·ln
.
a
d
1
(loga x) = x·ln
.
dx
a
This formula is valid for all x > 0.
Example. [log10 x]′ = x·ln1 10 = 0.434…
.
x
1
Example. [log2 x]′ = x·ln
= 1.443…
.
2
x
3. The Chain Rule and Log
Using the Chain Rule, we can solve this example:
[ln(x2 + 4)]′ = x21+4 · (x2 + 4)′ = x22x+4 .

x)
The Chain Rule also gives us: [ln(sin x)]′ = sin1 x · (sin x)′ = (sin
= cot x.
sin x
More generally, By applying the Chain Rule, we then obtain

(x)
[ln g(x)]′ = gg(x)
.

(x)
Some students choose to memorize [ln g(x)]′ = gg(x)
but this formula is really an immediate
consequence of the Chain Rule applied to the formula for the derivative of the natural log.
4. An application example.
Show that the slopes of the tangent lines to y = ln(ex +1) at (x, y) tend to one as x approaches
infinity.
dy
We need to illustrate that limx→∞ dx
= 1.
dy
d
(ex + 1)′
ex
=
ln(ex + 1) = x
= x
.
dx
dx
e +1
e +1
3
Thus
y = ln(ex + 1)
1
ex
1
= lim
x
x→∞ e + 1
x→∞ 1 + e−x
1
=
limx→∞ (1 + e−x )
1
1
=
= 1+0
= 1.
−x
1 + limx→∞ e
dy
= lim
lim dx
x→∞
Recall that |x| =
5. The derivative of ln absolute value of x.
{
x if x > 0
−x if x 0 the formula reduces to dx
ln x = x1 , which have already shown to be true.
When x
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