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Derivative Rules for Functions
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The following list provides the rules
for finding the derivatives of common functions. Both the specific
and the general rules are provided; the specific rule first and the
general rule second. The general rules are the direct application
of the chain rule.
The rules are provided first then
observations of each. Following the rules are examples.
Justification for the rules are advanced and not provided at this
level.
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The exponential function is
encountered often in mathematics, Calculus in particular, and is
used to model many physical processes. Its definition is such
that its basic derivative is simply itself.
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1) exponential function
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The logarithm function is elated to
the exponential function. It is not seen as often as
the exponential function. It is used to handle physical data such
as pH levels in acid/base levels of solutions (chemistry) and
decibel (dB) levels in sound since the range of data in each span
powers of 10 in their values. In
effect we graph the exponents of values of this data on
logarithmic
axis.
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2) logarithm function
(Recall that ln is the
notation we use for the logarithm base e, that is loge.)
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The sine
and cosine functions are trigonometric functions and are used to
model phenomenon that is repetitive, that is periodic, in nature.
The motion of a pendulum is periodic; the expansion and
contraction of a spring is periodic; the motion of a chain link
fence when pushed from one side and let go is periodic; sound and
light waves are periodic; the waves in a pond resulting from a
rock splashing into the pond are periodic; the motion of your head
up and down as you walk or run is periodic. The sine and cosine
functions can be written in terms of the exponential function.
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3) sine function
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4) cosine function
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Comments:
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1) in the general equation set f(x) = x.
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The derivative of ex
is just ex.
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2) The logarithm is defined for positive numbers only,
so in the general definition g(x) > 0.
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With this in mind, we can still talk about ALL positive
and negative x (this, of course, excludes 0) if we use the
absolute value function and set g(x) = |x|. When we do this we
have:
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The derivative of the logarithm of the absolute value of x is
just its reciprocal.
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Notice we dropped the absolute value in the result. The reason
for this is the logarithm function is symmetric across the Y axis
for |x|. From a geometric perspective, recall that the derivative
at x is the slope of the tangent to the curve at x. The slopes
for the tangents for ln(|x|) are negative for negative x.
Therefore we must remove the absolute value signs from 1/x.
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Note about bases:
This rule for the derivative of a logarithm assumes the base of
the logarithm is the natural number e. For other bases the
change of base rule must be applied first. For example, the
common logarithm in terms of the natural number e is: log10
(x) = ln(x) / ln(10)
The observation is also true for the exponential function. The
derivative of the exponential function assumes the base is e.
For other bases you must first write that exponential in terms
of e. 10x = e ln(10)
x
Examples 11 and 16 below illustrate these techniques.
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3, 4) First review the the
sine and cosine functions. The sine function in that
plot is the blue curve. At x = 0 the slope for sin(x) is at its
maximum and is the peak value of cos(x). As you follow x until
the sine curve reaches its peak, the tangent at that point has the
slope 0 which is the value of the cosine curve at that point. If
you continue this approach using the cosine curve you will see
that the slopes of the tangents to the cosine curve are the
negative values of the sine curve at those points.
The sine and cosine functions are the derivatives of each
other. Only note that a negative sign is introduced for the
derivative of the cosine function.
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Problems involving derivatives of
these special functions usually involve the chain rule and for
logarithms we normally use the properties of logarithms first then
proceed with the derivative rules.
Examples:
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1)
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In this example f(x) is kx.
We find the derivative of kx and multiply it by ekx.
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2)
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Here we change the radical exponent to a fractional exponent
then use the power rule. Recall that (½ – 1)
is -½.
Now we bring back the radical sign.
One could leave the radical in the denominator, but we probably
should rationalize this fraction (multiplying by
 )
and get:
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3)
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As you can see, you can write the derivative of the exponent
then multiply by the exponential.
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4)
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We need to use the product rule for derivatives. In this case
f(x) is x and g(x) is e-x.
Recall the product rule is
f(x) g'(x) + f '(x) g(x).
e-x is the common factor, so we factor it from both
terms to arrive at the answer.
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5)
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Since there is a single term in the denominator we can
simplify this fraction first.
Now apply the sum of functions rule.
Now apply the reciprocal rule.
The two negative signs produce a positive result.
Simplify.
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6)
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Unlike example 5 we do not have a single term in the
denominator. We cannot simplify this fraction.
We need to apply the quotient rule for derivatives first.
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Be very careful. If you make a mistake it will, in all
likelihood, be made in this first step.
The derivative of ex
± 1) using the sum
rule is just
(ex
± 0), that is, ex.)
Factor out ex
then combine the terms in the second factor. The ex
terms add to zero and the negative sign distributes across the -1
to give a +1 which adds to the first +1 to give +2.
Write the numerator in the customary way, constants first.
There is no need to carry out the square of the denominator.
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7)
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We can use the product rule for
derivatives pair at a time, but it's simpler to use the product
rule for exponents first. Recall that exponential factors of the
same base can be combined into one exponential by adding the
exponents.
Also note that (1/x) = x-1
so its derivative is -1x(-1 -1) = -x-2 =
-1/x2
This answer is fine as is, there is no “cleaner”
way of writing it.
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8)
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This is a direct application of the derivative rule for
logarithms.
Notice the absolute values are dropped in the answer as
explained above.
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9)
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This example applies logarithm rules in the first step prior to
applying the derivative rule for logarithms.
The absolute value signs are dropped since x2
is always positive.
ln(3) is constant so, remember, its derivative is zero.
Using the product rule logarithms we can split the logarithm
into two terms.
Now we use the sum of functions rule for derivatives
Using the general rule for the derivative of logarithms we
could have immediately written from the start:
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10)
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Here the quotient rule for logarithms is used first.
(We cannot simplify this fraction, so we must use the logarithm
rule first.)
We cannot simplify these logarithm terms we since we do not
have products in their arguments. Now we use the derivative of
the difference of two functions. to produce two derivative
expressions.
We now directly use the general rule for the derivative of a
logarithm.
Watch the two minus signs here shown in green.
Now we have an algebra problem to combine these fractions.
Technically this step is a fine answer. This result is the
derivative of the original logarithm function.
As a review of algebra we'll proceed to write this as a single
fraction. First we need the same denominators, so multiply and divide
each by the others.
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Simplify the numerator first.
Multiply
by -1/-1 to remove the leading negative signs in both the
numerator and the denominator.
Direct
application of the logarithm rule requires a similar amount of
algebra.
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11)
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We have the common logarithm, base 10.
First we use the change of base rule for logarithms to write
this as a base e logarithm.
Now realize that ln(10) is constant so we can bring it outside
the derivative.
Now we use the logarithm rule for derivatives.
It's a matter of habit to write expressions with logarithms
this way. ln(10)x would be fine but is not commonly done.
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12)
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We begin with the chain rule.
In this case we have f(g(x)) where
f(x) = sin and g(x) = kx
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13)
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Like the previous example we must use the chain rule.
f(x) = cos(x) and g(x) = (1-2x)
Simplify noting that the two negative signs become one positive
sign.
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14)
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We use the product rule here.
Simplify.
Finally we recognize the trigonometric formula for cos(2x).
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15)
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We do not have a rule (in this page) to find the derivative of
tan(x). But we recognize that the tangent function is the ratio
of the sine over the cosine functions. First write tan(x) in
terms of sin(x) and cos(x) and use the quotient rule for
derivatives.
Apply the derivative rule for the cosine and sine functions.
Simplify watching the double negative signs in the numerator.
Recognize the sum of the squares rule for the sine and cosine
functions.
Finally recognize that
sec(x) = 1/cos(x) and write this result in terms of sec(x).
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16)
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Before we can apply the exponent rule, the base must be 'e'.
Using the change of base rule for exponents, ab
= eln(a)b we have
xcos(x)
=
(eln(x))cos(x) = eln(x)cos(x)
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First apply the general exponent rule for derivatives.
Now apply the product rule for derivatives.
Write the exponential with its original base x.
Simplify.
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