Derivatives using Logarithmic Differentiation

In summary, the conversation is about how to use logarithmic differentiation to find the derivative of y=e^(x^x). The attempt at a solution involves taking the natural logarithm of both sides and using the properties of logarithms to simplify the expression. However, the person becomes stuck and is unsure of how to proceed. They suggest taking the logarithm again, but are unsure of how to go from xLNx to LNx+1.
  • #1
howsockgothap
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Homework Statement



Using logarithmic differentiation calculate the derivative of y=e^(x^x)



The Attempt at a Solution



y=e^x^x
LNy=LNe^x^x
LNy=x^xLNe
...
Stuck!

This seems to be the only way you can do it, but once I get to that part I'm not sure what else there is to do. I know lne=1, so does that mean the answer is y'=e^(x^x)*x^x
 
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  • #2
[tex]\ln(\ln(y))=\ln(x^{x}\ln(e))=\ln(x^{x})=x\ln(x)[/tex]

And:
[tex](\ln(\ln(y))'=\frac{y'}{y\ln(y)}=\ln(x)+1[/tex]
whereby follows:
[tex]y'=y\ln(y)(\ln(x)+1)=e^{x^{x}}x^{x}(\ln(x)+1)[/tex]
 
  • #3
Maybe you need to take the logarithm two times...
 
  • #4
I don't see how it goes from xLNx to LNx+1
 

1. What is logarithmic differentiation?

Logarithmic differentiation is a method used in calculus to find the derivative of a function that contains a logarithm. It involves taking the natural logarithm of both sides of the original function and then using properties of logarithms to simplify the expression before taking the derivative.

2. When should I use logarithmic differentiation?

Logarithmic differentiation is useful when the function contains both multiplication and division, making it difficult to find the derivative using traditional methods. It is also helpful when the function involves complex expressions such as exponents or logarithms.

3. What are the steps for using logarithmic differentiation?

The steps for using logarithmic differentiation are as follows:

  • Take the natural logarithm of both sides of the original function.
  • Use logarithmic properties to simplify the expression.
  • Take the derivative of both sides using the chain rule and product rule as needed.
  • Solve for the derivative of the original function.

4. Can logarithmic differentiation be used for any type of function?

Yes, logarithmic differentiation can be used for any type of function that contains a logarithm, as long as the function can be simplified using logarithmic properties. It is particularly useful for functions with complex expressions.

5. How does logarithmic differentiation compare to other methods for finding derivatives?

Logarithmic differentiation is a useful method for finding derivatives of complex functions, but it is not always the most efficient method. Other methods, such as the power rule or chain rule, may be more straightforward for simpler functions. It is important to choose the method that is best suited for the particular function being evaluated.

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