# Derivative Of Logarithmic And Exponential Functions Pdf

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- Exponentials and Logarithms
- Differentiation of Exponential and Logarithmic Functions
- Differentiation of Exponential and Logarithmic Functions

## Exponentials and Logarithms

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 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. 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 is the logarithm easier to do now that we know the derivative of the exponential function.

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## Differentiation of Exponential and Logarithmic Functions

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. Using the product rule, the chain rule and the derivative of the natural logarithm, we have. Necessary cookies are absolutely essential for the website to function properly.

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.

Summary The derivatives of the exponential and logarithm functions are computed. By the end of your studying, you should know: The derivative of e x. The derivative of ln x. How to write a x in terms of e x ; and to use this formula to compute the derivative of a x. How to write log to the base a of x in terms of ln x ; and to use this formula to compute the derivative of the general logarithm function. On-screen applet instructions: For any x, the red dot represents the difference quotient for the natural log at x for a given h. Find y'.

## Differentiation 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.

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

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.

We will solve different types of problems given in the Worksheet on H.

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Natural logarithm is the logarithm to the base. e e. Notation: Summary x x dx.