Derivative Rules for Functions


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Derivative Rules for Functions



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.


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.

1) exponential function





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.

2) logarithm function


(Recall that ln is the notation we use for the logarithm base e, that is loge.)










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.

3) sine function





4) cosine function




Comments:


1) in the general equation set f(x) = x.


The derivative of ex is just ex.










2) The logarithm is defined for positive numbers only, so in the general definition g(x) > 0.

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:




The derivative of the logarithm of the absolute value of x is just its reciprocal.




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.


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.




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.






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:


1)

In this example f(x) is kx.


We find the derivative of kx and multiply it by ekx.




2)



Here we change the radical exponent to a fractional exponent then use the power rule. Recall that (½ – 1) is .






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:




3)



As you can see, you can write the derivative of the exponent then multiply by the exponential.


4)


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.







5)

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.





6)

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.



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.

7)


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.



8)

This is a direct application of the derivative rule for logarithms.



Notice the absolute values are dropped in the answer as explained above.




9)

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:







10)




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.



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.







11)

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.

12)

We begin with the chain rule.


In this case we have f(g(x)) where

f(x) = sin and g(x) = kx







13)

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.



14)

We use the product rule here.






Simplify.

Finally we recognize the trigonometric formula for cos(2x).





15)




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).





16)



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)






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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