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Chain Rule: The General Logarithm Rule - Problem 2

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Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

You can use the chain rule to find the derivative of a composite function involving natural logs, as well. Recall that the derivative of ln(x) is 1/x. For example, say f(x)=ln(g(x)), where g(x) is some other function of x. By the chain rule, take the derivative of the "outside" function and multiply it by the derivative of the "inside" function. With the derivative of logarithmic functions, the outside function is the logarithm itself, and the inside function is what is inside the logarithm. So, f'(x)=1/g(x) * g'(x).

We’re differentiating functions that are compositions of natural log and some other function. And in the next example, I’m going to do a problem where a property of natural logs will come in very handy. This property; natural log of A/B equals natural log of A minus natural log of B. Now you don’t need this property but its going to make the problem a lot easier to solve. So let’s take a look.

It says differentiate h(x) equals ln of 9 minus x over 2x plus 3. So I’m going to use this property of logs I just introduced to break this function up into 2 pieces. The log of a quotient can be broken up into the log of the numerator minus the log of the denominator. That’s just from properties of logs. Now when I differentiate it I can differentiate it in two pieces.

Now the derivative of the log of a function is 1 over that function. 1 over 9 minus x times the derivative of the function. The derivative of 9 minus x, that’s -1, minus, and the derivative of this guy is 1 over 2x plus 3 times the derivative of 2x plus 3, which is 2. So just to simplify, I get -1 over 9 minus x, minus 2 over 2x plus 3.

Now whenever you have -1 over 9 minus x, something like this, you can multiply any fraction top and bottom by -1. It doesn’t change the value of the fraction because you are effectively multiplying by 1. But let’s say we awn tot get a positive numerator for some reason or we’d like to have the x in front in the denominator. We multiply the top and bottom by -1 and we get +1 on top. -9 plus x is the same as x minus 9. That’s a much nicer looking answer. That’s my answer; the derivative of ln of this complicated quotient is 1 over x minus 9 minus 2 over 2x plus 3.

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