Find dy/dx (10-x)y^2=x^3

Problem

(10−x)*y2=x3

Solution

1. Identify the equation as one requiring implicit differentiation because y is not isolated.

2. Apply the product rule to the left side of the equation, treating y as a function of x

d(10−x)/d(x)*y2+(10−x)d(y2)/d(x)=d(x3)/d(x)

3. Differentiate each term with respect to x using the chain rule for the term involving y

−1⋅y2+(10−x)⋅2*yd(y)/d(x)=3*x2

4. Isolate the term containing d(y)/d(x) by moving −y2 to the right side of the equation.

2*y*(10−x)d(y)/d(x)=3*x2+y2

5. Solve for d(y)/d(x) by dividing both sides by 2*y*(10−x)

d(y)/d(x)=(3*x2+y2)/(2*y*(10−x))

Final Answer

d(y)/d(x)=(3*x2+y2)/(2*y*(10−x))

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