Differentiation Formulas
In the first section of this chapter we saw the
definition of the derivative and we computed a couple of derivatives using
the definition. As we saw in those examples there was a fair amount of work
involved in computing the limits and the functions that we worked with were not
terribly complicated.
For more complex functions using the definition of the
derivative would be an almost impossible task. Luckily for us we won�t have to
use the definition terribly often. We will have to use it on occasion, however
we have a large collection of formulas and properties that we can use to
simplify our life considerably and will allow us to avoid using the definition
whenever possible.
We will introduce most of these formulas over the course of
the next several sections. We will start in this section with some of the basic
properties and formulas. We will give the properties and formulas in this
section in both �prime� notation and �fraction� notation.
Properties
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1)
OR
In other words, to
differentiate a sum or difference all we need to do is differentiate the
individual terms and then put them back together with the appropriate
signs. Note as well that this property is not limited to two functions.
See the
Proof of Various Derivative Formulas section of the Extras chapter
to see the proof of this property. It�s a very simple proof using the
definition of the derivative.
2)
OR
 ,
c is any number
In other words, we can
�factor� a multiplicative constant out of a derivative if we need to.
See the
Proof of Various Derivative Formulas section of the Extras chapter
to see the proof of this property. |
Note that we have not included formulas for the derivative
of products or quotients of two functions here. The derivative of a product or
quotient of two functions is not the product or quotient of the derivatives of
the individual pieces. We will take a look at these in the next section.
Next, let�s take a quick look at a couple of basic
�computation� formulas that will allow us to actually compute some derivatives.
Formulas
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1)
If
 then
OR
The derivative of a
constant is zero. See the
Proof of Various Derivative Formulas section of the Extras chapter
to see the proof of this formula.
2)
If
 then
OR
 ,
n is any number.
This formula is
sometimes called the power rule. All we are doing here is
bringing the original exponent down in front and multiplying and then
subtracting one from the original exponent.
Note as well that in
order to use this formula n must be a number, it can�t be a
variable. Also note that the base, the x, must be a variable, it
can�t be a number. It will be tempting in some later sections to misuse
the Power Rule when we run in some functions where the exponent isn�t a
number and/or the base isn�t a variable.
See the
Proof of Various Derivative Formulas section of the Extras chapter
to see the proof of this formula. There are actually three different
proofs in this section. The first two restrict the formula to n
being an integer because at this point that is all that we can do at
this point. The third proof is for the general rule, but does suppose
that you�ve read most of this chapter. |
These are the only properties and formulas that we�ll give
in this section. Let�s compute some derivatives using these properties.
Example 1
Differentiate each of the following functions.
(a)
(b)
(c)
(d)
(e)
Solution
(a)
In this case we have the sum and difference of four terms
and so we will differentiate each of the terms using the first property from
above and then put them back together with the proper sign. Also, for each term
with a multiplicative constant remember that all we need to do is �factor� the
constant out (using the second property) and then do the derivative.
Notice that in the third term the exponent was a one and so
upon subtracting 1 from the original exponent we get a new exponent of zero. Now
recall that
 .
Don�t forget to do any basic arithmetic that needs to be done such as any
multiplication and/or division in the coefficients.
(b)
The point of this problem is to make sure that you deal
with negative exponents correctly. Here is the derivative.
Make sure that you correctly deal with the exponents in
these cases, especially the negative exponents. It is an easy mistake to �go
the other way� when subtracting one off from a negative exponent and get
 instead
of the correct
 .
(c)
Now in this function the second term is not correctly set
up for us to use the power rule. The power rule requires that the term be a
variable to a power only and the term must be in the numerator. So, prior to
differentiating we first need to rewrite the second term into a form that we can
deal with.
Note that we left the 3 in the denominator and only moved
the variable up to the numerator. Remember that the only thing that gets an
exponent is the term that is immediately to the left of the exponent. If we�d
wanted the three to come up as well we�d have written,
so be careful with this! It�s a very common mistake to
bring the 3 up into the numerator as well at this stage.
Now that we�ve gotten the function rewritten into a proper
form that allows us to use the Power Rule we can differentiate the function.
Here is the derivative for this part.
(d)
All of the terms in this function have roots in them. In
order to use the power rule we need to first convert all the roots to fractional
exponents. Again, remember that the Power Rule requires us to have a variable
to a number and that it must be in the numerator of the term. Here is the
function written in �proper� form.
In the last two terms we combined the exponents. You
should always do this with this kind of term. In a later section we will learn
of a technique that would allow us to differentiate this term without combining
exponents, however it will take significantly more work to do. Also don�t
forget to move the term in the denominator of the third term up to the
numerator. We can now differentiate the function.
Make sure that you can deal with fractional exponents. You
will see a lot of them in this class.
(e)
In all of the previous examples the exponents have been
nice integers or fractions. That is usually what we�ll see in this class.
However, the exponent only needs to be a number so don�t get excited about
problems like this one. They work exactly the same.
The answer is a little messy and we won�t reduce the
exponents down to decimals. However, this problem is not terribly difficult it
just looks that way initially.
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