# Derivatives of implicit realtionships.

#### Petrus

##### Well-known member
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
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
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

$\chi$ $\sigma$