Solve 2y/y^2-4/frac{3{y^2-4y+4}}

$$\frac { \frac { 2 y } { y ^ { 2 } - 4 } } { \frac { 3 } { y ^ { 2 } - 4 y + 4 } }$$

Evaluate

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

Short Solution Steps

Divide \frac{2y}{y^{2}-4} by \frac{3}{y^{2}-4y+4} by multiplying \frac{2y}{y^{2}-4} by the reciprocal of \frac{3}{y^{2}-4y+4}.

$$\frac{2y\left(y^{2}-4y+4\right)}{\left(y^{2}-4\right)\times 3}$$

Factor the expressions that are not already factored.

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

Cancel out y-2 in both numerator and denominator.

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

Expand the expression.

$$\frac{2y^{2}-4y}{3y+6}$$

Differentiate w.r.t. y

$$\frac{2\left(y^{2}+4y-4\right)}{3\left(y+2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Divide \frac{2y}{y^{2}-4} by \frac{3}{y^{2}-4y+4} by multiplying \frac{2y}{y^{2}-4} by the reciprocal of \frac{3}{y^{2}-4y+4}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{2y\left(y^{2}-4y+4\right)}{\left(y^{2}-4\right)\times 3})$$

Factor the expressions that are not already factored in \frac{2y\left(y^{2}-4y+4\right)}{\left(y^{2}-4\right)\times 3}.

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

Cancel out y-2 in both numerator and denominator.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{2y^{2}-4y}{3\left(y+2\right)})$$

Use the distributive property to multiply 3 by y+2.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{2y^{2}-4y}{3y+6})$$

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.

$$\frac{\left(3y^{1}+6\right)\frac{\mathrm{d}}{\mathrm{d}y}(2y^{2}-4y^{1})-\left(2y^{2}-4y^{1}\right)\frac{\mathrm{d}}{\mathrm{d}y}(3y^{1}+6)}{\left(3y^{1}+6\right)^{2}}$$

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}.

$$\frac{\left(3y^{1}+6\right)\left(2\times 2y^{2-1}-4y^{1-1}\right)-\left(2y^{2}-4y^{1}\right)\times 3y^{1-1}}{\left(3y^{1}+6\right)^{2}}$$

Simplify.

$$\frac{\left(3y^{1}+6\right)\left(4y^{1}-4y^{0}\right)-\left(2y^{2}-4y^{1}\right)\times 3y^{0}}{\left(3y^{1}+6\right)^{2}}$$

Multiply 3y^{1}+6 times 4y^{1}-4y^{0}.

$$\frac{3y^{1}\times 4y^{1}+3y^{1}\left(-4\right)y^{0}+6\times 4y^{1}+6\left(-4\right)y^{0}-\left(2y^{2}-4y^{1}\right)\times 3y^{0}}{\left(3y^{1}+6\right)^{2}}$$

Multiply 2y^{2}-4y^{1} times 3y^{0}.

$$\frac{3y^{1}\times 4y^{1}+3y^{1}\left(-4\right)y^{0}+6\times 4y^{1}+6\left(-4\right)y^{0}-\left(2y^{2}\times 3y^{0}-4y^{1}\times 3y^{0}\right)}{\left(3y^{1}+6\right)^{2}}$$

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

$$\frac{3\times 4y^{1+1}+3\left(-4\right)y^{1}+6\times 4y^{1}+6\left(-4\right)y^{0}-\left(2\times 3y^{2}-4\times 3y^{1}\right)}{\left(3y^{1}+6\right)^{2}}$$

Simplify.

$$\frac{12y^{2}-12y^{1}+24y^{1}-24y^{0}-\left(6y^{2}-12y^{1}\right)}{\left(3y^{1}+6\right)^{2}}$$

Combine like terms.

$$\frac{6y^{2}+24y^{1}-24y^{0}}{\left(3y^{1}+6\right)^{2}}$$

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

$$\frac{6y^{2}+24y-24y^{0}}{\left(3y+6\right)^{2}}$$

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

$$\frac{6y^{2}+24y-24}{\left(3y+6\right)^{2}}$$