Find dy/dx x^(2/3)+y^(2/3)=5

Problem

x(2/3)+y(2/3)=5

Solution

1. Differentiate both sides of the equation with respect to x using the power rule and the chain rule for the term involving y

d(x(2/3))/d(x)+d(y(2/3))/d(x)=d(5)/d(x)

2. Apply the power rule to the x term and the chain rule to the y term.

2/3*x(−1/3)+2/3*y(−1/3)d(y)/d(x)=0

3. Isolate the term containing d(y)/d(x) by subtracting 2/3*x(−1/3) from both sides.

2/3*y(−1/3)d(y)/d(x)=−2/3*x(−1/3)

4. Solve for d(y)/d(x) by dividing both sides by 2/3*y(−1/3)

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

5. Simplify the expression by canceling the common factor 2/3 and moving the negative exponents to the opposite parts of the fraction.

d(y)/d(x)=−(y(1/3))/(x(1/3))

Final Answer

d(y)/d(x)=−√(3,y/x)

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