Question

Find d2y/dx2 in terms of x and y. y^4 = x^7

Answer 1

`y^4=x^7`

Differentiating once gives


`4y^3(\mathrm dy)/(\mathrm dx)=7x^6`

Differentiating again gives


`12y^2\left((\mathrm dy)/(\mathrm dx)\right)^2+4y^3(\mathrm d^2y)/(\mathrm dx^2)=42x^5`

From the first result, you get


`4y^3(\mathrm dy)/(\mathrm dx)=7x^6\implies(\mathrm dy)/(\mathrm dx)=(7x^6)/(4y^3)`

and plugging this into the second gives


`12y^2\left((7x^6)/(4y^3)\right)^2+4y^3(\mathrm d^2y)/(\mathrm dx^2)=42x^5`

`12y^2(49x^(12))/(16y^6)+4y^3(\mathrm d^2y)/(\mathrm dx^2)=42x^5`

`(147x^(12))/(4y^4)+4y^3(\mathrm d^2y)/(\mathrm dx^2)=42x^5`

and solving for
`(\mathrm d^2y)/(\mathrm dx^2)` gives


`(\mathrm d^2y)/(\mathrm dx^2)=(42x^5-(147x^(12))/(4y^4))/(4y^3)`

`(\mathrm d^2y)/(\mathrm dx^2)=(168x^5y^4-147x^(12))/(16y^7)`