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:
$$D_x(x^ny^m)=D_x((x+y)^{n+m})$$
Applying the product rule in the left side and the chain rule in the right side:
$$nx^{n-1}y^m+my^{m-1}x^nD_xy=(n+m)(x+y)^{n+m-1}(1+D_xy)$$

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
If I implicitly differentiate it, I get:
$$y + xD_xy = 2(x+y)(1+D_xy)$$,
which, after some manipulation, becomes:
$$D_xy = \frac{2x+y}{-x-2y}$$
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:
$$D_x(x^ny^m)=D_x((x+y)^{n+m})$$
Applying the product rule in the left side and the chain rule in the right side:
$$nx^{n-1}y+my^{m-1}x=(n+m)(x+y)^{n+m-1}D_xy$$

That last symbol should be $D_x(x+ y)$, not $D_xy$. Also you have dropped the exponents on $x^n$ and $y^m$ 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)$$Solve that for D_xy.<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I tried to do many manipulations, but I don't see any way to get the expected result.<br /> <br /> Could the equation given be wrong? I tried to let n = 1 and m = 1 and see what happens:<br /> x¹y¹ = (x+y)¹⁺¹<br /> xy = (x+y)²<br /> This result seems to suggest that the equation given is not correct. Or am I doing something wrong?<br /> <br /> Thank you in advance. </div> </div> </blockquote>$$[/tex]

 

[pc2-brazil]

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