Find dy/dx x^2-2xy=6

Problem

x2−2*x*y=6

Solution

1. Differentiate both sides of the equation with respect to x treating y as a function of x

d(x2)/d(x)−(d(2)*x*y)/d(x)=d(6)/d(x)

2. Apply the power rule to the first term and the product rule to the second term.

2*x−(2*y+2*xd(y)/d(x))=0

3. Distribute the negative sign through the parentheses.

2*x−2*y−2*xd(y)/d(x)=0

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

−2*xd(y)/d(x)=2*y−2*x

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

d(y)/d(x)=(2*y−2*x)/(−2*x)

6. Simplify the fraction by dividing the numerator and denominator by −2

d(y)/d(x)=(x−y)/x

Final Answer

d(y)/d(x)=(x−y)/x

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