Find the Derivative - d/dx -(x^2)/(y^2)

Problem

d()/d(x)−(x2)/(y2)

Solution

1. Identify the expression as a quotient where y is an implicit function of x

2. Apply the quotient rule to the expression (x2)/(y2) keeping the negative sign outside.

3. Differentiate the numerator x2 with respect to x to get 2*x

4. Differentiate the denominator y2 with respect to x using the chain rule to get 2*yd(y)/d(x)

5. Substitute these into the quotient rule formula (vd(u)/d(x)−ud(v)/d(x))/(v2)

−(y2d(x2)/d(x)−x2d(y2)/d(x))/((y2)2)

6. Simplify the resulting expression by performing the differentiation.

−(y2*(2*x)−x2*(2*yd(y)/d(x)))/(y4)

7. Distribute the negative sign and simplify the fraction by dividing terms by y

(2*x2*yd(y)/d(x)−2*x*y2)/(y4)

(2*x2d(y)/d(x)−2*x*y)/(y3)

Final Answer

d()/d(x)−(x2)/(y2)=(2*x2d(y)/d(x)−2*x*y)/(y3)

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