Find dy/dx x^2=(4x^2y^3+1)^2

Problem

d()/d(x)*[x2=(4*x2*y3+1)2]

Solution

1. Apply implicit differentiation by taking the derivative of both sides of the equation with respect to x

d(x2)/d(x)=d(4*x2*y3+1)/d(x)

2. Differentiate the left side using the power rule.

2*x=d(4*x2*y3+1)/d(x)

3. Apply the chain rule to the right side, treating (4*x2*y3+1) as the inner function.

2*x=2*(4*x2*y3+1)⋅d(4*x2*y3+1)/d(x)

4. Differentiate the inner expression using the product rule on the term 4*x2*y3

2*x=2*(4*x2*y3+1)⋅(8*x*y3+12*x2*y2d(y)/d(x))

5. Simplify the equation by dividing both sides by 2.

x=(4*x2*y3+1)*(8*x*y3+12*x2*y2d(y)/d(x))

6. Distribute the terms on the right side to isolate the derivative.

x=32*x3*y6+48*x4*y5d(y)/d(x)+8*x*y3+12*x2*y2d(y)/d(x)

7. Group terms containing d(y)/d(x) on one side and move all other terms to the opposite side.

x−32*x3*y6−8*x*y3=(48*x4*y5+12*x2*y2)d(y)/d(x)

8. Factor out the derivative d(y)/d(x)

x*(1−32*x2*y6−8*y3)=12*x2*y2*(4*x2*y3+1)d(y)/d(x)

9. Solve for dy/dx by dividing by the coefficient of the derivative.

d(y)/d(x)=(x*(1−32*x2*y6−8*y3))/(12*x2*y2*(4*x2*y3+1))

10. Simplify the fraction by canceling the common factor x

d(y)/d(x)=(1−32*x2*y6−8*y3)/(12*x*y2*(4*x2*y3+1))

Final Answer

d(y)/d(x)=(1−32*x2*y6−8*y3)/(48*x3*y5+12*x*y2)

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