Find dy/dx 1=3x+2x^2y^2

Problem

1=3*x+2*x2*y2

Solution

1. Differentiate both sides of the equation with respect to x treating y as a function of x

d(1)/d(x)=d(3*x+2*x2*y2)/d(x)

2. Apply the sum rule and the constant rule on the left side.

0=(d(3)*x)/d(x)+(d(2)*x2*y2)/d(x)

3. Apply the product rule and the chain rule to the term 2*x2*y2

0=3+(2*x2⋅d(y2)/d(x)+y2⋅(d(2)*x2)/d(x))

4. Compute the derivatives, using the chain rule for d(y2)/d(x)=2*yd(y)/d(x)

0=3+2*x2*(2*yd(y)/d(x))+y2*(4*x)

5. Simplify the expression.

0=3+4*x2*yd(y)/d(x)+4*x*y2

6. Isolate the term containing d(y)/d(x) by moving all other terms to the left side.

−3−4*x*y2=4*x2*yd(y)/d(x)

7. Solve for d(y)/d(x) by dividing both sides by 4*x2*y

d(y)/d(x)=(−3−4*x*y2)/(4*x2*y)

Final Answer

d(y)/d(x)=−(3+4*x*y2)/(4*x2*y)

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