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

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In summary, the conversation is about finding the derivative of y=x^y. One method shown is taking the natural logarithm of both sides and rearranging the equation to solve for y'. However, the person asking the question was looking for a different form of the solution. It is also noted that in order to express y in terms of x, x must first be expressed in terms of y.
  • #1
fffbone
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How do you find the derivative of y=x^y ?
 
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  • #2
[tex]\begin{align*}
y&=x^y \\
\ln y&=y\ln x \\
\frac{y^\prime}{y}&=y^\prime\ln x+\frac{y}{x} \\
y^\prime\left(\frac{1}{y}-\ln x\right)&=\frac{y}{x} \\
y^\prime&=\frac{y^2}{x-xy\ln x}
\end{align*}[/tex]
 
  • #3
Not quite the answer I was looking for, but thanks any how. I already know how the problem is solved.
 
  • #4
You asked me how to find the derivative. I showed you a way to do it. What were you looking for?
 
  • #5
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.
 
  • #6
Usually people want the derivative of y wrt x. Plus, you gave an equation for y in terms of x and y.
 
  • #7
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).
 

1. What is the definition of a derivative?

A derivative is a mathematical concept that measures the rate of change of a function with respect to its independent variable. It represents the slope of a tangent line to a curve at a given point.

2. How do you calculate a derivative?

The derivative of a function can be calculated by using the limit definition of a derivative, which involves taking the limit of the difference quotient as the change in the independent variable approaches zero. Alternatively, certain rules and formulas can be applied to calculate the derivative of a given function.

3. What is the difference between a derivative and an antiderivative?

A derivative is a measure of the rate of change of a function, while an antiderivative is the reverse process of finding a function whose derivative is equal to the original function. In other words, a derivative is a function's slope, while an antiderivative is the original function itself.

4. Why are derivatives important in science and engineering?

Derivatives are important in science and engineering because they allow us to analyze the behavior and relationships between variables in mathematical models. They are used to calculate rates of change, find maximum and minimum values, and optimize functions in various fields, such as physics, economics, and engineering.

5. Are there any real-life applications of derivatives?

Yes, derivatives have many real-life applications. They are used in calculating velocity and acceleration in physics, determining marginal profit and cost in economics, and optimizing production processes in engineering. They are also used in finance to calculate the risk and return of investments.

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