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So far, we have learned how to differentiate a variety of functions, including trigonometric, inverse, and implicit functions. In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs , exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions.

## Derivatives of Exponential and Logarithmic Functions

The next set of functions that we want to take a look at are exponential and logarithm functions. We will take a more general approach however and look at the general exponential and logarithm function. We want to differentiate this. We can therefore factor this out of the limit. This gives,. Therefore, the derivative becomes,. Here are three of them. So, this definition leads to the following fact,. Eventually we will be able to show that for a general exponential function we have,. In this case we will need to start with the following fact about functions that are inverses of each other.

So, how is this fact useful to us? Well recall that the natural exponential function and the natural logarithm function are inverses of each other and we know what the derivative of the natural exponential function is!

It can also be shown that,. In this case, unlike the exponential function case, we can actually find the derivative of the general logarithm function. All that we need is the derivative of the natural logarithm, which we just found, and the change of base formula.

Using the change of base formula we can write a general logarithm as,. Putting all this together gives,. In later sections as we get more formulas under our belt they will become more complicated. First, we will need the derivative. We need this to determine if the object ever stops moving since at that point provided there is one the velocity will be zero and recall that the derivative of the position function is the velocity of the object.

The two derivatives are,. It is important to note that with the Power rule the exponent MUST be a constant and the base MUST be a variable while we need exactly the opposite for the derivative of an exponential function. It is easy to get locked into one of these formulas and just use it for both of these. Notes Quick Nav Download.

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If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Example 1 Differentiate each of the following functions. Not much to this one. Just remember to use the product rule on the second term.

Show Solution First, we will need the derivative. ## 3.9: Derivatives of Exponential and Logarithmic Functions

The next set of functions that we want to take a look at are exponential and logarithm functions. We will take a more general approach however and look at the general exponential and logarithm function. We want to differentiate this. We can therefore factor this out of the limit. This gives,. ## Differentiation of Exponential and Logarithmic Functions

Marsden and A. Weinstein pp Cite as. Unable to display preview.

As with the sine function, we don't know anything about derivatives that allows us to compute the derivatives of the exponential and logarithmic functions without going back to basics. Let's do a little work with the definition again:. Yes it does, but we will prove this property at the end of this section. We can look at some examples.

So far, we have learned how to differentiate a variety of functions, including trigonometric, inverse, and implicit functions. In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs , exponential functions play an important role in modeling population growth and the decay of radioactive materials.

So far, we have learned how to differentiate a variety of functions, including trigonometric, inverse, and implicit functions.

### Exponentials and Logarithms

Сьюзан Флетчер нетерпеливо мерила шагами туалетную комнату шифровалки и медленно считала от одного до пятидесяти. Голова у нее раскалывалась. Еще немного, - повторяла она мысленно.

Код ошибки 22. Она попыталась вспомнить, что это. Сбои техники в Третьем узле были такой редкостью, что номера ошибок в ее памяти не задерживалось. #### Derivative of the Exponential Function

Завтра они скажут мне спасибо, - подумал он, так и не решив, правильно ли поступает. Набрав полные легкие воздуха, Чатрукьян открыл металлический шкафчик старшего сотрудника лаборатории систем безопасности. На полке с компьютерными деталями, спрятанными за накопителем носителей информации, лежала кружка выпускника Стэнфордского университета и тестер. Не коснувшись краев, он вытащил из нее ключ Медеко. - Поразительно, - пробурчал он, - что сотрудникам лаборатории систем безопасности ничего об этом не известно. ГЛАВА 47 - Шифр ценой в миллиард долларов? - усмехнулась Мидж, столкнувшись с Бринкерхоффом в коридоре. Д-дэвид… - Сьюзан не знала, что за спиной у нее собралось тридцать семь человек. Сьюзан, я люблю.  - Слова лились потоком, словно ждали много лет, чтобы сорваться с его губ.  - Я люблю. Я люблю . ## Bruxtingpiwal

d dx. (loge x) = 1 x. We can use these results and the rules that we have learnt already to differentiate functions which involve exponentials or logarithms. Example.

## Dan E.

3 Use logarithmic differentiation to determine the derivative of a function. So far, we have learned how to differentiate a variety of functions, including trigonometric.