# Implicit Differentiation

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If 2x^2 + 3xy = 12x + 5y, what is dy/dx?

I put the x's on one side and got:

2x^2 - 12 x = 3xy +5y

then 4x - 12 = 3y + 5, but I'm not sure how to get a dy and dx.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

2x^2 + 3xy = 12x + 5y

To solve for dy/dx, you have to use implicit differentiation. Take d/dx of each side:

d/dx (2x^2 + 3xy) = d/dx (12x + 5y)

d/dx (2x^2) + d/dx (3xy) = d/dx (12x) + d/dx (5y)

Taking derivatives:

4x + 3(d/dx (xy)) = 12 + 5dy/dx

Using the chain rule:

4x + 3(xdy/dx + y) = 12 + 5dy/dx

4x + 3xdy/dx + 3y = 12 + 5dy/dx

Subtract 12:

4x + 3xdy/dx + 3y - 12 = 5dy/dx

Subtract 3xdy/dx:

4x + 3y - 12 = 5dy/dx - 3xdy/dx

Factor:

4x + 3y - 12 = (5 - 3x)dy/dx

Divide:

dy/dx = (4x + 3y + 12)/(5 - 3x)

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