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2.6. DERIVATIVES OF INVERSE FUNCTIONS
Finally, we recall that since g(x) = ln(x), e g(x) = e ln(x) = x, and thus
g
′ (x) =
1
x
.
Natural Logarithm: For all positive real numbers x,
d
dx
[ln(x)] =
1
x
.
This rule for the natural logarithm function now joins our list of other basic derivative
rules that we have already established. There are two particularly interesting things to
note about the fact that
d
dx [ln(x)] =
1
x . One is that this rule is restricted to only apply
to positive values of x, as these are the only values for which the original function is
defined. The other is that for the first time in our work, differentiating a basic function
of a particular type has led to a function of a very different nature: the derivative of the
natural logarithm is not another logarithm, nor even an exponential function, but rather a
rational one.
Derivatives of logarithms may now be computed in concert with all of the rules known
to date. For instance, if f (t) = ln(t 2 + 1), then by the chain rule, f ′ (t) =
1
t 2 +1
· 2t.
Activity 2.16.
For each function given below, find its derivative.
(a) h(x) = x 2 ln(x)
(b) p(t) =
ln(t)
e t + 1
(c) s(y) = ln(cos(y) + 2)
(d) z(x) = tan(ln(x))
(e) m(z) = ln(ln(z))
⊳
In addition to the important rule we have derived for the derivative of the natural
log functions, there are additional interesting connections to note between the graphs of
f (x) = e x and f −1 (x) = ln(x).
In Figure 2.9, we are reminded that since the natural exponential function has the
property that its derivative is itself, the slope of the tangent to y = e x is equal to the
height of the curve at that point. For instance, at the point A = (ln(0.5), 0.5), the slope
of the tangent line is m A = 0.5, and at B = (ln(5), 5), the tangent line’s slope is m B = 5.
At the corresponding points A ′ and B ′ on the graph of the natural logarithm function
(which come from reflecting across the line y = x), we know that the slope of the tangent
line is the reciprocal of the x-coordinate of the point (since
d
dx [ln(x)] =
1
x ). Thus, with
A ′ = (0.5, ln(0.5)), we have m A ′ =
1
0.5 = 2, and at B ′ = (5, ln(5)), m B ′ =
1
5 .
2.6. DERIVATIVES OF INVERSE FUNCTIONS
Finally, we recall that since g(x) = ln(x), e g(x) = e ln(x) = x, and thus
g
′ (x) =
1
x
.
Natural Logarithm: For all positive real numbers x,
d
dx
[ln(x)] =
1
x
.
This rule for the natural logarithm function now joins our list of other basic derivative
rules that we have already established. There are two particularly interesting things to
note about the fact that
d
dx [ln(x)] =
1
x . One is that this rule is restricted to only apply
to positive values of x, as these are the only values for which the original function is
defined. The other is that for the first time in our work, differentiating a basic function
of a particular type has led to a function of a very different nature: the derivative of the
natural logarithm is not another logarithm, nor even an exponential function, but rather a
rational one.
Derivatives of logarithms may now be computed in concert with all of the rules known
to date. For instance, if f (t) = ln(t 2 + 1), then by the chain rule, f ′ (t) =
1
t 2 +1
· 2t.
Activity 2.16.
For each function given below, find its derivative.
(a) h(x) = x 2 ln(x)
(b) p(t) =
ln(t)
e t + 1
(c) s(y) = ln(cos(y) + 2)
(d) z(x) = tan(ln(x))
(e) m(z) = ln(ln(z))
⊳
In addition to the important rule we have derived for the derivative of the natural
log functions, there are additional interesting connections to note between the graphs of
f (x) = e x and f −1 (x) = ln(x).
In Figure 2.9, we are reminded that since the natural exponential function has the
property that its derivative is itself, the slope of the tangent to y = e x is equal to the
height of the curve at that point. For instance, at the point A = (ln(0.5), 0.5), the slope
of the tangent line is m A = 0.5, and at B = (ln(5), 5), the tangent line’s slope is m B = 5.
At the corresponding points A ′ and B ′ on the graph of the natural logarithm function
(which come from reflecting across the line y = x), we know that the slope of the tangent
line is the reciprocal of the x-coordinate of the point (since
d
dx [ln(x)] =
1
x ). Thus, with
A ′ = (0.5, ln(0.5)), we have m A ′ =
1
0.5 = 2, and at B ′ = (5, ln(5)), m B ′ =
1
5 .