Find the Derivative - d/dx -(4x^2)/(5y^2)

Problem

d()/d(x)−(4*x2)/(5*y2)

Solution

1. Identify the expression as a quotient involving two variables, where y is assumed to be a function of x requiring the use of implicit differentiation.

2. Apply the constant multiple rule by pulling the constant factor −4/5 outside of the derivative.

d()/d(x)−(4*x2)/(5*y2)=−4/5d()/d(x)(x2)/(y2)

3. Apply the quotient rule, which states d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2) where u=x2 and v=y2

d(x2)/d(x)=2*x

d(y2)/d(x)=2*yd(y)/d(x)

4. Substitute these derivatives into the quotient rule formula.

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

5. Simplify the expression by factoring out 2*y from the numerator and reducing the power of y in the denominator.

(2*x*y2−2*x2*yd(y)/d(x))/(y4)=(2*x*y−2*x2d(y)/d(x))/(y3)

6. Combine the constant factor back into the simplified derivative.

−4/5(2*x*y−2*x2d(y)/d(x))/(y3)=(−8*x*y+8*x2d(y)/d(x))/(5*y3)

Final Answer

d()/d(x)−(4*x2)/(5*y2)=(8*x2d(y)/d(x)−8*x*y)/(5*y3)

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