Solve frac{x^{2}y^3}{2x^2y-2x^2y^2}=

$$\frac { x ^ { 2 } y ^ { 3 } } { 2 x ^ { 2 } y - 2 x ^ { 2 } y ^ { 2 } } =$$

Evaluate

$$\frac{y^{2}}{2\left(1-y\right)}$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{x^{2}y^{3}}{2y\left(-y+1\right)x^{2}}$$

Cancel out yx^{2} in both numerator and denominator.

$$\frac{y^{2}}{2\left(-y+1\right)}$$

Expand the expression.

$$\frac{y^{2}}{-2y+2}$$

Differentiate w.r.t. y

$$\frac{y\left(2-y\right)}{2\left(1-y\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Factor the expressions that are not already factored in \frac{x^{2}y^{3}}{2x^{2}y-2x^{2}y^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{x^{2}y^{3}}{2y\left(-y+1\right)x^{2}})$$

Cancel out yx^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{2}}{2\left(-y+1\right)})$$

Use the distributive property to multiply 2 by -y+1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{2}}{-2y+2})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(-2y^{1}+2\right)\frac{\mathrm{d}}{\mathrm{d}y}(y^{2})-y^{2}\frac{\mathrm{d}}{\mathrm{d}y}(-2y^{1}+2)}{\left(-2y^{1}+2\right)^{2}}$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{\left(-2y^{1}+2\right)\times 2y^{2-1}-y^{2}\left(-2\right)y^{1-1}}{\left(-2y^{1}+2\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(-2y^{1}+2\right)\times 2y^{1}-y^{2}\left(-2\right)y^{0}}{\left(-2y^{1}+2\right)^{2}}$$

Expand using distributive property.

$$\frac{-2y^{1}\times 2y^{1}+2\times 2y^{1}-y^{2}\left(-2\right)y^{0}}{\left(-2y^{1}+2\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{-2\times 2y^{1+1}+2\times 2y^{1}-\left(-2y^{2}\right)}{\left(-2y^{1}+2\right)^{2}}$$

Do the arithmetic.

$$\frac{-4y^{2}+4y^{1}-\left(-2y^{2}\right)}{\left(-2y^{1}+2\right)^{2}}$$

Combine like terms.

$$\frac{\left(-4-\left(-2\right)\right)y^{2}+4y^{1}}{\left(-2y^{1}+2\right)^{2}}$$

Subtract -2 from -4.

$$\frac{-2y^{2}+4y^{1}}{\left(-2y^{1}+2\right)^{2}}$$

Factor out 2y.

$$\frac{2y\left(-y^{1}+2y^{0}\right)}{\left(-2y^{1}+2\right)^{2}}$$

For any term t, t^{1}=t.

$$\frac{2y\left(-y+2y^{0}\right)}{\left(-2y+2\right)^{2}}$$

For any term t except 0, t^{0}=1.

$$\frac{2y\left(-y+2\times 1\right)}{\left(-2y+2\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{2y\left(-y+2\right)}{\left(-2y+2\right)^{2}}$$