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# Derivatives Of Logarithmic And Exponential Functions Examples Pdf

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We can now use derivatives of logarithmic and exponential functions to solve various types of problems eg. Cessna taking off. A Cessna plane takes off from an airport at sea level and its altitude in feet at time t in minutes is given by. At low altitudes, where the air is more dense, the rate of climb is good, but as you go higher, the rate decreases. Note: In aviation, height above sea level is meaured in feet.

## 7. Applications: Derivatives of Logarithmic and Exponential Functions

The derivative of ln x. The derivative of e with a functional exponent. The derivative of ln u x. The general power rule. In the next Lesson , we will see that e is approximately 2. The system of natural logarithms is in contrast to the system of common logarithms , which has 10 as its base and is used for most practical work.

Now we consider the logarithmic function with arbitrary base and obtain a formula for its derivative. Then the last relation can be rewritten as. Here we used the property of the limit of a composite function given that the logarithmic function is continuous. Differentiate using the quotient rule :. Using the product and difference rules, we have.

As with the sine, 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. Yes it does, but we will not prove this fact. We can look at some examples. As we can already see, some of these limits will be less than 1 and some larger than 1. What about the logarithm function? This too is hard, but as the cosine function was easier to do once the sine was done, so the logarithm is easier to do now that we know the derivative of the exponential function.

## Derivatives of Exponential Functions

The derivative of a logarithmic function is the reciprocal of the argument. As always, the chain rule tells us to also multiply by the derivative of the argument. Differentiate by taking the reciprocal of the argument. Don't forget the chain rule! Differentiate using the formula for derivatives of logarithmic functions. When the argument of the logarithmic function involves products or quotients we can use the properties of logarithms to make differentiating easier. It's derivative is.

## Exponentials and Logarithms

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.

### DERIVATIVES OF LOGARITHMIC

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.

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. Just as when we found the derivatives of other functions, we can find the derivatives of exponential and logarithmic functions using formulas. As we develop these formulas, we need to make certain basic assumptions. The proofs that these assumptions hold are beyond the scope of this course.

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.

#### Exercises 4.7

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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. Just as when we found the derivatives of other functions, we can find the derivatives of exponential and logarithmic functions using formulas.

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