Please help me with this, given that x^2y^2-3xy+4xy^3=5 find dy/dx by implicit differentiation
just remember to use the product rule. Also, dx/dx = 1
x^2y^2-3xy+4xy^3=5
2xy^2+2x^2yy' - 3y-3xy' + 4y^3+12xy^2y' = 0
y'(2x^2y-3x+12xy^2) = -2xy^2+3y-4y^3
y' =
-2xy^2+3y-4y^3
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2x^2y-3x+12xy^2
To find the derivative dy/dx using implicit differentiation, we need to differentiate both sides of the equation with respect to x. Here's how you can do that step-by-step:
Step 1: Take the derivative of both sides with respect to x.
d/dx(x^2y^2) - d/dx(3xy) + d/dx(4xy^3) = d/dx(5)
Step 2: Apply the product rule for derivatives.
Use the product rule to differentiate each term:
d/dx(x^2y^2) = 2xy^2(dx/dx) + x^2(2y)(dy/dx)
d/dx(3xy) = 3y(dx/dx) + 3x(dy/dx)
d/dx(4xy^3) = 4y^3(dx/dx) + 4x(dy/dx)(3y^2)
We can simplify a bit:
2xy^2 + 2x^2y(dy/dx) - 3y + 3x(dy/dx) + 4y^3(dx/dx) + 12xy^2(dy/dx) = 0
Step 3: Combine like terms.
Rearrange the equation to gather the terms with dy/dx on one side:
(2x^2y + 3x + 12xy^2) (dy/dx) = - (2xy^2 + 3y + 4y^3)
Step 4: Solve for dy/dx.
Now, we can isolate dy/dx by dividing both sides of the equation by (2x^2y + 3x + 12xy^2):
dy/dx = - (2xy^2 + 3y + 4y^3) / (2x^2y + 3x + 12xy^2)
So, the derivative dy/dx is equal to - (2xy^2 + 3y + 4y^3) divided by (2x^2y + 3x + 12xy^2).