Find dy/dx (11x+3y)^(1/3)=x^2

Problem

(11*x+3*y)(1/3)=x2

Solution

1. Differentiate both sides with respect to x using the chain rule on the left side and the power rule on the right side.

d(11*x+3*y)/d(x)=d(x2)/d(x)

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

1/3*(11*x+3*y)(−2/3)⋅d(11*x+3*y)/d(x)=2*x

3. Differentiate the inner function (11*x+3*y) with respect to x

1/3*(11*x+3*y)(−2/3)⋅(11+3d(y)/d(x))=2*x

4. Isolate the term containing the derivative by multiplying both sides by 3*(11*x+3*y)(2/3)

11+3d(y)/d(x)=6*x*(11*x+3*y)(2/3)

5. Subtract 11 from both sides to begin solving for d(y)/d(x)

3d(y)/d(x)=6*x*(11*x+3*y)(2/3)−11

6. Divide by 3 to find the final expression for the derivative.

d(y)/d(x)=(6*x*(11*x+3*y)(2/3)−11)/3

Final Answer

d(y)/d(x)=2*x*(11*x+3*y)(2/3)−11/3

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