implicit differentianion- the algebra of it

**bosmith** · October 24th 2009, 06:11 AM

I hope I can get some help understanding the math of this problem. The problem is 2x^(2/3) + y^(2/3). I did this: d/dx (y^2/3) = -d/dx ((-2x^(2/3). I come up with dy/dx = -2x^(2/3)/y^(2/3). My graphing claculator says the answer is -2y^1/3)/x^(1/3). Can someone please help me understand the math that comes up with the solution on my calculator, or am I just doing this wrong?

**tonio** · October 24th 2009, 06:27 AM

Originally Posted by bosmith

I hope I can get some help understanding the math of this problem. The problem is 2x^(2/3) + y^(2/3). I did this: d/dx (y^2/3) = -d/dx ((-2x^(2/3). I come up with dy/dx = -2x^(2/3)/y^(2/3). My graphing claculator says the answer is -2y^1/3)/x^(1/3). Can someone please help me understand the math that comes up with the solution on my calculator, or am I just doing this wrong?

The problem is ill-posed, I think: what is $2x^{2\slash3}+y^{2\slash3}$ equal to?

Assuming the above is equal to a constant, then deriving implicitly:

$\frac{4}{3}x^{-1\slash3}dx+\frac{2}{3}y^{-1\slash3}dy=0\Longrightarrow \frac{dy}{dx}=-2\left(\frac{y}{x}\right)^{1\slash3}$

I'm guessing you took inadvertently $\frac{dx}{dy}$ or perhaps you overlooked the minus signs in the powers.

Tonio

**Scott H** · October 24th 2009, 06:39 AM

To find the derivative of $x^n$ , we must subtract $1$ from the exponent in addition to multiplying by $n$ . This gives us

$\begin{aligned} 2x^{2/3}+y^{2/3}&=0\\ \frac{d}{dx}(2x^{2/3}+y^{2/3})&=0\\ \frac{d}{dx}(2x^{2/3})+\frac{d}{dx}(y^{2/3})&=0\\ 2\cdot\frac{2}{3}x^{-1/3}+\frac{2}{3}y^{-1/3}\frac{dy}{dx}&=0\\ \frac{dy}{dx}&=\left(-\frac{4}{3}x^{-1/3}\right)\left(\frac{2}{3}y^{-1/3}\right)^{-1}=-\frac{2x^{-1/3}}{y^{-1/3}}=-\frac{2y^{1/3}}{x^{1/3}}=-2\sqrt[3]{\frac{y}{x}}. \end{aligned}$

**bosmith** · October 24th 2009, 06:56 AM

Thank you both very much. I have been studying Calculus I, so much I sometimes forget the basics. The question came from a sample problem from my Instructor. There was no equal sign in the problem, somewhat puzzling. Doesn't matter though because your assistance put me on track. I simply forgot to subtract (-1) from the exponents. Once you showed me that, I now understand what I failed to do. Thanks.

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This is a grateful thanks to Tonio and Scott H

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