Solve 1/2y-3-18y/8y^3-27

$$\frac { 1 } { 2 y - 3 } - \frac { 18 y } { 8 y ^ { 3 } - 27 }$$

Evaluate

$$\frac{2y-3}{4y^{2}+6y+9}$$

Short Solution Steps

Factor 8y^{3}-27.

$$\frac{1}{2y-3}-\frac{18y}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2y-3 and \left(2y-3\right)\left(4y^{2}+6y+9\right) is \left(2y-3\right)\left(4y^{2}+6y+9\right). Multiply \frac{1}{2y-3} times \frac{4y^{2}+6y+9}{4y^{2}+6y+9}.

$$\frac{4y^{2}+6y+9}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}-\frac{18y}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}$$

Since \frac{4y^{2}+6y+9}{\left(2y-3\right)\left(4y^{2}+6y+9\right)} and \frac{18y}{\left(2y-3\right)\left(4y^{2}+6y+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4y^{2}+6y+9-18y}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}$$

Combine like terms in 4y^{2}+6y+9-18y.

$$\frac{4y^{2}-12y+9}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}$$

Factor the expressions that are not already factored in \frac{4y^{2}-12y+9}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}.

$$\frac{\left(2y-3\right)^{2}}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}$$

Cancel out 2y-3 in both numerator and denominator.

$$\frac{2y-3}{4y^{2}+6y+9}$$

Differentiate w.r.t. y

$$-\frac{2\left(\left(2y-3\right)^{2}-27\right)}{\left(4y^{2}+6y+9\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Factor 8y^{3}-27.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{2y-3}-\frac{18y}{\left(2y-3\right)\left(4y^{2}+6y+9\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2y-3 and \left(2y-3\right)\left(4y^{2}+6y+9\right) is \left(2y-3\right)\left(4y^{2}+6y+9\right). Multiply \frac{1}{2y-3} times \frac{4y^{2}+6y+9}{4y^{2}+6y+9}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{4y^{2}+6y+9}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}-\frac{18y}{\left(2y-3\right)\left(4y^{2}+6y+9\right)})$$

Since \frac{4y^{2}+6y+9}{\left(2y-3\right)\left(4y^{2}+6y+9\right)} and \frac{18y}{\left(2y-3\right)\left(4y^{2}+6y+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{4y^{2}+6y+9-18y}{\left(2y-3\right)\left(4y^{2}+6y+9\right)})$$

Combine like terms in 4y^{2}+6y+9-18y.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{4y^{2}-12y+9}{\left(2y-3\right)\left(4y^{2}+6y+9\right)})$$

Factor the expressions that are not already factored in \frac{4y^{2}-12y+9}{\left(2y-3\right)\left(4y^{2}+6y+9\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\left(2y-3\right)^{2}}{\left(2y-3\right)\left(4y^{2}+6y+9\right)})$$

Cancel out 2y-3 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{2y-3}{4y^{2}+6y+9})$$

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.

$$\frac{\left(4y^{2}+6y^{1}+9\right)\frac{\mathrm{d}}{\mathrm{d}y}(2y^{1}-3)-\left(2y^{1}-3\right)\frac{\mathrm{d}}{\mathrm{d}y}(4y^{2}+6y^{1}+9)}{\left(4y^{2}+6y^{1}+9\right)^{2}}$$

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}.

$$\frac{\left(4y^{2}+6y^{1}+9\right)\times 2y^{1-1}-\left(2y^{1}-3\right)\left(2\times 4y^{2-1}+6y^{1-1}\right)}{\left(4y^{2}+6y^{1}+9\right)^{2}}$$

Simplify.

$$\frac{\left(4y^{2}+6y^{1}+9\right)\times 2y^{0}-\left(2y^{1}-3\right)\left(8y^{1}+6y^{0}\right)}{\left(4y^{2}+6y^{1}+9\right)^{2}}$$

Multiply 4y^{2}+6y^{1}+9 times 2y^{0}.

$$\frac{4y^{2}\times 2y^{0}+6y^{1}\times 2y^{0}+9\times 2y^{0}-\left(2y^{1}-3\right)\left(8y^{1}+6y^{0}\right)}{\left(4y^{2}+6y^{1}+9\right)^{2}}$$

Multiply 2y^{1}-3 times 8y^{1}+6y^{0}.

$$\frac{4y^{2}\times 2y^{0}+6y^{1}\times 2y^{0}+9\times 2y^{0}-\left(2y^{1}\times 8y^{1}+2y^{1}\times 6y^{0}-3\times 8y^{1}-3\times 6y^{0}\right)}{\left(4y^{2}+6y^{1}+9\right)^{2}}$$

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

$$\frac{4\times 2y^{2}+6\times 2y^{1}+9\times 2y^{0}-\left(2\times 8y^{1+1}+2\times 6y^{1}-3\times 8y^{1}-3\times 6y^{0}\right)}{\left(4y^{2}+6y^{1}+9\right)^{2}}$$

Simplify.

$$\frac{8y^{2}+12y^{1}+18y^{0}-\left(16y^{2}+12y^{1}-24y^{1}-18y^{0}\right)}{\left(4y^{2}+6y^{1}+9\right)^{2}}$$

Combine like terms.

$$\frac{-8y^{2}+24y^{1}+36y^{0}}{\left(4y^{2}+6y^{1}+9\right)^{2}}$$

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

$$\frac{-8y^{2}+24y+36y^{0}}{\left(4y^{2}+6y+9\right)^{2}}$$

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

$$\frac{-8y^{2}+24y+36\times 1}{\left(4y^{2}+6y+9\right)^{2}}$$

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

$$\frac{-8y^{2}+24y+36}{\left(4y^{2}+6y+9\right)^{2}}$$