Find dy/dx 3y^2=(4x-5)/(4x+5)

Problem

3*y2=(4*x−5)/(4*x+5)

Solution

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

(d(3)*y2)/d(x)=d()/d(x)(4*x−5)/(4*x+5)

2. Apply the chain rule to the left side to find the derivative in terms of y and d(y)/d(x)

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

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

6*yd(y)/d(x)=(16*x+20−16*x+20)/((4*x+5)2)

6*yd(y)/d(x)=40/((4*x+5)2)

5. Isolate the derivative d(y)/d(x) by dividing both sides by 6*y

d(y)/d(x)=40/(6*y*(4*x+5)2)

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

d(y)/d(x)=20/(3*y*(4*x+5)2)

Final Answer

d(y)/d(x)=20/(3*y*(4*x+5)2)

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