Calculus problem involving implicit differentiation.

[pc2-brazil]

Good afternoon,

This is not actually a homework question; it's for self-study. I'm reading a Calculus book, and one of its exercises asks the following:
If xⁿy^m = (x+y)^n+m, show that xD_xy = y (where D_xy is the derivative of y with respect to x).

The only way I could think of to get the correct result is by implicit differentiation. I tried to do implicit differentiation of the given equation, but it got me nowhere:
[tex]D_x(x^ny^m)=D_x((x+y)^{n+m})[/tex]
Applying the product rule in the left side and the chain rule in the right side:
[tex]nx^{n-1}y^m+my^{m-1}x^nD_xy=(n+m)(x+y)^{n+m-1}(1+D_xy)[/tex]

I tried to do many manipulations, but I don't see any way to get the expected result.

Could the equation given be wrong? I tried to let n = 1 and m = 1 and see what happens:
x¹y¹ = (x+y)¹⁺¹
xy = (x+y)²
If I implicitly differentiate it, I get:
[tex]y + xD_xy = 2(x+y)(1+D_xy)[/tex],
which, after some manipulation, becomes:
[tex]D_xy = \frac{2x+y}{-x-2y}[/tex]
This result seems to suggest that the equation given is not correct. Or am I doing something wrong?

Thank you in advance.

 

[HallsofIvy]

Good afternoon,

This is not actually a homework question; it's for self-study. I'm reading a Calculus book, and one of its exercises asks the following:
If xⁿy^m = (x+y)^n+m, show that xD_xy = y (where D_xy is the derivative of y with respect to x).

The only way I could think of to get the correct result is by implicit differentiation. I tried to do implicit differentiation of the given equation, but it got me nowhere:
[tex]D_x(x^ny^m)=D_x((x+y)^{n+m})[/tex]
Applying the product rule in the left side and the chain rule in the right side:
[tex]nx^{n-1}y+my^{m-1}x=(n+m)(x+y)^{n+m-1}D_xy[/tex]

That last symbol should be [itex]D_x(x+ y)[/itex], not [itex]D_xy[/itex]. Also you have dropped the exponents on [itex]x^n[/itex] and [itex]y^m[/itex] where they were not differentiated. That is, you should have
[tex]nx^{n-1}y^m+ mx^ny^{m-1}D_xy= (n+m)(x+ y)^{n+ m- 1}(1+ D_xy)[tex]
Solve that for D_xy.

I tried to do many manipulations, but I don't see any way to get the expected result.

Could the equation given be wrong? I tried to let n = 1 and m = 1 and see what happens:
x¹y¹ = (x+y)¹⁺¹
xy = (x+y)²
This result seems to suggest that the equation given is not correct. Or am I doing something wrong?

Thank you in advance.

 

[pc2-brazil]

I made these typing mistakes while writing the TeX expression.
When I solve for D_xy, I find:
[tex]D_xy = \frac{(n+m)(x+y)^{n+m-1}-nx^{n-1}y^m}{my^{m-1}x^n-(n+m)(x+y)^{n+m-1}}[/tex]
But I'm not very sure on how I should continue.
Thank you in advance.