To make life easy, we will break this question up into parts.

Part A:

Find the derivative with respect to x of: y4

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

Part B:

Find the derivative with respect to x of 2x2y2

Now to find the derivative of 2x2y2 with respect to x we must recognise that it is a product.

If we let u = 2x2 and v = y2 then we have:

`d/(dx)(2x^2y^2)=u(dv)/(dx)+v(du)/(dx)`

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

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

Part C:

Now

`d/(dx)6x^2=12x`

and

`d/(dx)(7)=0`

Now to find `(dy)/(dx)` for the whole expression:

`y^4+2x^2y^2+6x^2=7`

Working left to right, using our answers from above:

`[4y^3(dy)/(dx)]+[4x^2y(dy)/(dx)+4xy^2]+` `[12x]=0`

This gives us, on collecting terms:

`(4y^3+4x^2y)(dy)/(dx)=-4xy^2-12x`

So we have the required expression:

`(dy)/(dx)=(-4xy^2-12x)/(4y^3+4x^2y)=(-xy^2-3x)/(y^3+x^2y)`

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