Small problem understanding application of chain rule

In summary, the conversation discusses the correct proofs for the derivative of ln|x| and the general rule for the derivative of ln(u). It also mentions the derivative of |x| and how it can be used to find the derivative of ln(|x|). The speaker is confused about their work and asks for help finding their mistake.
  • #1
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Homework Statement


I have proven in two ways (correctly) that the derivative of ln|x| = 1/x (note absolute value does vanish)

Now I open my textbook and see a general rule that [tex]\frac{d}{dx} ln (u) = \frac{u'}{u}[/tex]

And the not so general derivative of |x| is [tex] \frac{d}{dx} |x| = \frac{x}{|x|}[/tex]

So using these statements [tex]\frac{d}{dx} ln (|x|) = \frac{|x|'}{|x|}[/tex]
[tex]=\frac{(\frac{x}{|x|})}{|x|}[/tex]
[tex]=\frac{x}{|x|^{2}}[/tex]

I've looked over this a few times and I can't see what I've done wrong. I mean I'm looking for a silly mistake but I don't see it. Can you see where I've gone wrong? Whats going on here?
 
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  • #2
abs(x)^2 = x^2
 

1. What is the chain rule and why is it important?

The chain rule is a calculus rule that allows us to find the derivative of a composite function. It is important because many real-world problems involve composite functions, and the chain rule allows us to find the rate of change in these situations.

2. How do I know when to use the chain rule?

You should use the chain rule when you have a function that is composed of two or more other functions. For example, if you have a function f(x) = g(h(x)), you would need to use the chain rule to find the derivative of f(x).

3. Can you explain the concept of "inner" and "outer" functions in the chain rule?

The "outer" function is the main function that is being applied to the input, while the "inner" function is the function being applied to the output of the outer function. In the example f(x) = g(h(x)), h(x) is the inner function and g(x) is the outer function.

4. How do I apply the chain rule in practical situations?

To apply the chain rule, you will need to identify the outer and inner functions in your composite function. Then, you can use the formula (f(g(x)))' = f'(g(x)) * g'(x) to find the derivative. Be sure to substitute in the derivative of the inner function and the derivative of the outer function.

5. Are there any common mistakes when using the chain rule?

One common mistake is forgetting to substitute in the derivative of the inner function and the derivative of the outer function. It is also important to correctly identify the inner and outer functions and use the correct notation when applying the chain rule. It is helpful to practice with different examples to avoid these mistakes.

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