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3.6Derivatives of Logarithmic FunctionsRecall how to differentiate inverse functions using implicit differentiation. Since the natural logarithm is the inverse function of the natural exponential,

How to fill out derivatives of logarithmic functions

How to fill out derivatives of logarithmic functions:

1. Understand the basic rules of logarithmic differentiation. These rules include the logarithmic differentiation rule, which states that the derivative of ln(x) with respect to x is 1/x, and the chain rule, which allows you to differentiate composite functions.
2. Identify the logarithmic function and any surrounding functions that may require the chain rule. For example, if you have the function f(x) = ln(x^2), you would need to differentiate both ln(x) and x^2 separately.
3. Apply the logarithmic differentiation rule to the logarithmic function. For example, if you have the function f(x) = ln(x), the derivative would be f'(x) = 1/x.
4. Apply the chain rule, if necessary, to the surrounding functions. Continuing with the previous example, if you have the function f(x) = ln(x^2), you would differentiate ln(x) as 1/x and differentiate x^2 as 2x, then multiply the two derivatives together to get the final result: f'(x) = 2x/x = 2.
5. Simplify the derivative if possible. This may involve combining like terms or canceling out common factors.
6. Check your derivative by plugging in different values of x and comparing it to the original function. This step helps to ensure accuracy and identify any potential errors.

Who needs derivatives of logarithmic functions?

1. Students and researchers in the field of mathematics and calculus often need to use derivatives of logarithmic functions. These concepts are fundamental in understanding the behavior of logarithmic equations and their rates of change.
2. Engineers and scientists who deal with exponential growth or decay models, such as population growth or radioactive decay, rely on derivatives of logarithmic functions to analyze and predict the behavior of these systems.
3. Financial analysts and economists may use derivatives of logarithmic functions in various financial models, such as compound interest calculations, where logarithmic differentiation helps in determining growth rates and optimizing investments.

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