Question

Find dy/dx by implicit differentiation: x^2y^3-7xy^2=10

Answer 1

`x^2y^3-7xy^2=10`

Differentiate both sides wrt
`x`:

`(\mathrm d(x^2y^3-7xy^2))/(\mathrm dx)=(\mathrm d(10))/(\mathrm dx)`

`(\mathrm d(x^2y^3))/(\mathrm dx)-7(\mathrm d(xy^2))/(\mathrm dx)=0`

By the product rule,

`\left(y^3(\mathrm d(x^2))/(\mathrm dx)+x^2(\mathrm d(y^3))/(\mathrm dx)\right)-7\left(y^2(\mathrm d(x))/(\mathrm dx)+x(\mathrm d(y^2))/(\mathrm dx)\right)=0`

By the power and chain rules,

`\left(y^3(2x)+x^2\left(3y^2(\mathrm dy)/(\mathrm dx)\right)\right)-7\left(y^2+x\left(2y(\mathrm dy)/(\mathrm dx)\right)\right)=0`

`\left(2xy^3+3x^2y^2(\mathrm dy)/(\mathrm dx)\right)-7\left(y^2+2xy(\mathrm dy)/(\mathrm dx)\right)=0`

`(3x^2y^2-14xy)(\mathrm dy)/(\mathrm dx)=7y^2-2xy^3`

`\implies(\mathrm dy)/(\mathrm dx)=(7y^2-2xy^3)/(3x^2y^2-14xy)`

and if
`y\\eq0`,

`\implies\boxed{(\mathrm dy)/(\mathrm dx)=(7y-2xy^2)/(3x^2y-14x)}`