Question

How do you use implicit differentiation to find `(dy)/(dx)` given `4x^2=2y^3+4y`?

Answer 1

I found:
`(dy)/(dx)=(4x)/(3y^2+2)`

Explanation:

We differentiate as usual BUT remembering to include the term `(dy)/(dx)` when we differentiate any `y`; so we get:
`8x=6y^2(dy)/(dx)+4(dy)/(dx)`
Collect `(dy)/(dx)`:
`(dy)/(dx)=(8x)/(6y^2+4)=(8x)/(2(3y^2+2))=(4x)/(3y^2+2)`

Answer 2

Please see the explanation.

Explanation:

For the x term, you can just use the power rule:

`(d(4x^2))/dx = 8x`

For the y terms, you use the power rule but then multiply by `dy/dx`

`(d(2y^3 + 4y))/dx = (6y^2 + 4)dy/dx`

NOTE: You are really using the chain rule along with the power rule.

Put the equation back together:

`8x = (6y^2 + 4)dy/dx`

Solve for `dy/dx`

`dy/dx = (8x)/(6y^2 + 4)`

Remove `2/2`:

`dy/dx = (4x)/(3y^2 + 2)`

Done.