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Derivatives of implicit realtionships.

Petrus

Well-known member
Feb 21, 2013
739
Hello,
Today did me and my friend talked of derivate and he asked about some help. Then he asked me is it possible to derivate $y^3+5x^2=5x-2y$ and i was clueless how i derivate that. Is this difficoult to derivate?is it possible to do it?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Re: Derivate of functions.

Hello,
Today did me and my friend talked of derivate and he asked about some help. Then he asked me is it possible to derivate $y^3+5x^2=5x-2y$ and i was clueless how i derivate that. Is this difficoult to derivate?is it possible to do it?
You need to use the chain rule. This says that if $z$ is a function of $y$ and $y$ is a function of $x$ then $\dfrac{dz}{dx} = \dfrac{dz}{dy}\dfrac{dy}{dx}.$ So for example if you want to differentiate $y^3$ with respect to $x$, the chain rule says that $\dfrac{d}{dx}(y^3) = 3y^2\dfrac{dy}{dx}.$ You can then differentiate both sides of the equation $y^3+5x^2=5x-2y$ with respect to $x$, to get $3y^2\frac{dy}{dx}+ 10x = 5 -2\frac{dy}{dx}.$ Now solve that equation to find an expression for $\frac{dy}{dx}.$

That process is called implicit differentiation.
 

chisigma

Well-known member
Feb 13, 2012
1,704
Re: Derivate of functions.

Hello,
Today did me and my friend talked of derivate and he asked about some help. Then he asked me is it possible to derivate $y^3+5x^2=5x-2y$ and i was clueless how i derivate that. Is this difficoult to derivate?is it possible to do it?
If You write the equation as...

$\displaystyle f(x,y)= y^{3} + 2\ y + 5\ x^{2} - 5\ x =0$ (1)

... You obtain the implicit definition of a function $y= \varphi(x)$. In the XVIII century the Italian mathematician and senator of the Kingdom Ulisse Dini demonstrated that, under appropriate conditions, the derivative of that function can be obtained as...


$\displaystyle \varphi^{\ '}(x)= - \frac{f^{\ '}_{x}(x,y)}{f^{\ '}_{y}(x,y)}$ (2)


Kind regards


$\chi$ $\sigma$