How to Find the Derivative of y=x^y When y is Not Expressed in terms of x?

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SUMMARY

The derivative of the equation y = x^y, where y is not expressed in terms of x, can be found using implicit differentiation. The process involves taking the natural logarithm of both sides, leading to the equation ln y = y ln x. The final derivative is expressed as y' = (y^2) / (x - xy ln x). An alternative expression for the derivative is x' = y^(1/y) * (1/y^2 - ln(y)/y^2), which is more aligned with the user's expectations. This discussion highlights the complexity of differentiating equations where variables are interdependent.

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  • Understanding of implicit differentiation
  • Familiarity with logarithmic properties
  • Knowledge of derivatives and their notation
  • Basic algebraic manipulation skills
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  • Study implicit differentiation techniques in calculus
  • Learn about logarithmic differentiation methods
  • Explore the properties of exponential functions
  • Investigate the implications of expressing variables in terms of each other
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Students and professionals in mathematics, particularly those studying calculus and differential equations, as well as educators seeking to enhance their understanding of implicit differentiation techniques.

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How do you find the derivative of y=x^y ?
 
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\begin{align*}<br /> y&amp;=x^y \\<br /> \ln y&amp;=y\ln x \\<br /> \frac{y^\prime}{y}&amp;=y^\prime\ln x+\frac{y}{x} \\<br /> y^\prime\left(\frac{1}{y}-\ln x\right)&amp;=\frac{y}{x} \\<br /> y^\prime&amp;=\frac{y^2}{x-xy\ln x}<br /> \end{align*}
 
Not quite the answer I was looking for, but thanks any how. I already know how the problem is solved.
 
You asked me how to find the derivative. I showed you a way to do it. What were you looking for?
 
I was looking for something more like this: x'=y^(1/y)*(1/y^2-ln(y)/y^2).

I solved it though, guess just got stuck for a minute.
 
Usually people want the derivative of y wrt x. Plus, you gave an equation for y in terms of x and y.
 
Since y does not equal f(x), y can not be expressed in terms of x. We would have to express x in terms of y.

Of course, it would be much better if the equation was x=y^x, then it can be expressed as y=x^(1/x).
 

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