Implicit Differentiation Shortcut

[Jimmy25]

Homework Statement

differentiate:

y^x = y / sqrt(x² + y²)

Homework Equations

The Attempt at a Solution

I solved this problem by taking the ln of both sides and then solving. It seems from the context of the problem set that this was supposed to be easier than that. Am I missing a simple shortcut to the solution or did I do it the only way that is possible?

 

[LCKurtz]

Nothing wrong with taking the ln first. You could also just differentiate it implicitly as it stands, using the quotient rule on the right. Either way should work if you slog through the details.

 

[Jimmy25]

But how would implicitly differentiate y^x?

 

[Whitishcube]

think of it as a function inside of another function. how do we differentiate functions that have a function inside of them?

 

[Jimmy25]

If it were a function inside another function you could use the chain rule. But in this case the variable y is being raised to the power of x another variable. If it were a number such as e raised to the power of a function you would take the derivative of e^function and then take the derivative of the function. But in this case how to take the derivative of function^function without taking a ln?

 

[LCKurtz]

If it were a function inside another function you could use the chain rule. But in this case the variable y is being raised to the power of x another variable. If it were a number such as e raised to the power of a function you would take the derivative of e^function and then take the derivative of the function. But in this case how to take the derivative of function^function without taking a ln?

When you different a function f(x,y) with respect to x with y an implicit function of x:

$$\frac d {dx} f(x,y) = f_x + f_y y'$$

For f(x,y) = y^x this gives y^xln(y) + xy^x-1y'.