MATH 155 SUPPLEMENTAL NOTES 2

LOGARITHMIC DIFFERENTIATION

EXAMPLE 1:

Well, we let us find the derivative by using brute force.

SOLUTION 1A:

Then simplify the above, if you like. That was not that bad, but there are functions in which the derivative could become more complicated. To handle the above example and more complicated functions; we will learn another way finding the derivative of the function. The method that we will be discussing will be called logarithmic differentiation. So, let us use this method to find the derivative of the function stated in example 1.

SOLUTION 1B:

Here are the steps that will be needed for logarithmic differentiation applied to this function.

Step 1: Take the natural logs of both sides of the function.

Step 2: Use the properties of logarithms to simplify the right hand side of the equation.

Step 3: Differentiate both sides of the equation.

Step 4: Solve for dy/dx.

Step 5: Substitute in for y.

Do not forget to substitute in for y. This is a common error when using this method.

Which solution is correct? Both are!! Simplify both to prove it to yourself!

Which one is method is best for some complex function? Logarithmic differentiation is. Logarithmic differentiation takes the natural log of a complex function, then applies the properties of logarithms to simplify the expression. By simplifying the expression, the derivatives become straightforward. I always use logarithmic differentiation when I am dealing with functions that are composed of radicals, products of factors, or quotients of factors.

EXAMPLE 2:

SOLUTION:

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EXAMPLE 3:

SOLUTION:

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EXAMPLE 4:

SOLUTION:

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If you follow the five steps that I outlined when doing logarithmic differentiation, you should not have any problems. Notice that logarithmic differentiation simplifies a complicated quotient, product, or radical function. It is a tool. Know when to use it, and when not to. If the function is straight forward, such as polynomial, do straight differentiation. Work through these examples. If you have any questions on this topic, feel free to ask questions.

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