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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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