Solve 6y-2+y/9y^2-1

$$6 y - 2 + \frac { y } { 9 y ^ { 2 } - 1 }$$

Evaluate

$$\frac{54y^{3}-18y^{2}-5y+2}{9y^{2}-1}$$

Short Solution Steps

Factor 9y^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 6y-2 times \frac{\left(3y-1\right)\left(3y+1\right)}{\left(3y-1\right)\left(3y+1\right)}.

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

Since \frac{\left(6y-2\right)\left(3y-1\right)\left(3y+1\right)}{\left(3y-1\right)\left(3y+1\right)} and \frac{y}{\left(3y-1\right)\left(3y+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(6y-2\right)\left(3y-1\right)\left(3y+1\right)+y.

$$\frac{54y^{3}-6y-18y^{2}+2+y}{\left(3y-1\right)\left(3y+1\right)}$$

Combine like terms in 54y^{3}-6y-18y^{2}+2+y.

$$\frac{54y^{3}-5y-18y^{2}+2}{\left(3y-1\right)\left(3y+1\right)}$$

Expand \left(3y-1\right)\left(3y+1\right).

$$\frac{54y^{3}-5y-18y^{2}+2}{9y^{2}-1}$$

Differentiate w.r.t. y

$$\frac{486y^{4}-117y^{2}+5}{\left(9y^{2}-1\right)^{2}}$$