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

Problem

3*y2=(2*x−5)/(2*x+5)

Solution

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

(d(3)*y2)/d(x)=d()/d(x)(2*x−5)/(2*x+5)

2. Apply the chain rule to the left side to differentiate the term involving y

6*yd(y)/d(x)=d()/d(x)(2*x−5)/(2*x+5)

3. Apply the quotient rule to the right side using the formula (vd(u)/d(x)−ud(v)/d(x))/(v2)

6*yd(y)/d(x)=((2*x+5)*(2)−(2*x−5)*(2))/((2*x+5)2)

4. Simplify the numerator on the right side by distributing and combining like terms.

6*yd(y)/d(x)=(4*x+10−4*x+10)/((2*x+5)2)

6*yd(y)/d(x)=20/((2*x+5)2)

5. Isolate the derivative by dividing both sides by 6*y

d(y)/d(x)=20/(6*y*(2*x+5)2)

6. Reduce the fraction by dividing the numerator and the coefficient in the denominator by their greatest common divisor, which is 2

d(y)/d(x)=10/(3*y*(2*x+5)2)

Final Answer

d(y)/d(x)=10/(3*y*(2*x+5)2)

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