Solve 5y/y^2-7y-4/2y-14+9/y

$$\frac { 5 y } { y ^ { 2 } - 7 y } - \frac { 4 } { 2 y - 14 } + \frac { 9 } { y }$$

Evaluate

$$\frac{3\left(4y-21\right)}{y\left(y-7\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{5y}{y^{2}-7y}.

$$\frac{5y}{y\left(y-7\right)}-\frac{4}{2y-14}+\frac{9}{y}$$

Cancel out y in both numerator and denominator.

$$\frac{5}{y-7}-\frac{4}{2y-14}+\frac{9}{y}$$

Factor 2y-14.

$$\frac{5}{y-7}-\frac{4}{2\left(y-7\right)}+\frac{9}{y}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y-7 and 2\left(y-7\right) is 2\left(y-7\right). Multiply \frac{5}{y-7} times \frac{2}{2}.

$$\frac{5\times 2}{2\left(y-7\right)}-\frac{4}{2\left(y-7\right)}+\frac{9}{y}$$

Since \frac{5\times 2}{2\left(y-7\right)} and \frac{4}{2\left(y-7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\times 2-4}{2\left(y-7\right)}+\frac{9}{y}$$

Do the multiplications in 5\times 2-4.

$$\frac{10-4}{2\left(y-7\right)}+\frac{9}{y}$$

Do the calculations in 10-4.

$$\frac{6}{2\left(y-7\right)}+\frac{9}{y}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(y-7\right) and y is 2y\left(y-7\right). Multiply \frac{6}{2\left(y-7\right)} times \frac{y}{y}. Multiply \frac{9}{y} times \frac{2\left(y-7\right)}{2\left(y-7\right)}.

$$\frac{6y}{2y\left(y-7\right)}+\frac{9\times 2\left(y-7\right)}{2y\left(y-7\right)}$$

Since \frac{6y}{2y\left(y-7\right)} and \frac{9\times 2\left(y-7\right)}{2y\left(y-7\right)} have the same denominator, add them by adding their numerators.

$$\frac{6y+9\times 2\left(y-7\right)}{2y\left(y-7\right)}$$

Do the multiplications in 6y+9\times 2\left(y-7\right).

$$\frac{6y+18y-126}{2y\left(y-7\right)}$$

Combine like terms in 6y+18y-126.

$$\frac{24y-126}{2y\left(y-7\right)}$$

Factor the expressions that are not already factored in \frac{24y-126}{2y\left(y-7\right)}.

$$\frac{6\left(4y-21\right)}{2y\left(y-7\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{3\left(4y-21\right)}{y\left(y-7\right)}$$

Expand y\left(y-7\right).

$$\frac{3\left(4y-21\right)}{y^{2}-7y}$$

Use the distributive property to multiply 3 by 4y-21.

$$\frac{12y-63}{y^{2}-7y}$$

Differentiate w.r.t. y

$$\frac{3\left(-4y^{2}+42y-147\right)}{\left(y\left(y-7\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Factor the expressions that are not already factored in \frac{5y}{y^{2}-7y}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5y}{y\left(y-7\right)}-\frac{4}{2y-14}+\frac{9}{y})$$

Cancel out y in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5}{y-7}-\frac{4}{2y-14}+\frac{9}{y})$$

Factor 2y-14.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5}{y-7}-\frac{4}{2\left(y-7\right)}+\frac{9}{y})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y-7 and 2\left(y-7\right) is 2\left(y-7\right). Multiply \frac{5}{y-7} times \frac{2}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5\times 2}{2\left(y-7\right)}-\frac{4}{2\left(y-7\right)}+\frac{9}{y})$$

Since \frac{5\times 2}{2\left(y-7\right)} and \frac{4}{2\left(y-7\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{5\times 2-4}{2\left(y-7\right)}+\frac{9}{y})$$

Do the multiplications in 5\times 2-4.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{10-4}{2\left(y-7\right)}+\frac{9}{y})$$

Do the calculations in 10-4.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{6}{2\left(y-7\right)}+\frac{9}{y})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(y-7\right) and y is 2y\left(y-7\right). Multiply \frac{6}{2\left(y-7\right)} times \frac{y}{y}. Multiply \frac{9}{y} times \frac{2\left(y-7\right)}{2\left(y-7\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{6y}{2y\left(y-7\right)}+\frac{9\times 2\left(y-7\right)}{2y\left(y-7\right)})$$

Since \frac{6y}{2y\left(y-7\right)} and \frac{9\times 2\left(y-7\right)}{2y\left(y-7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{6y+9\times 2\left(y-7\right)}{2y\left(y-7\right)})$$

Do the multiplications in 6y+9\times 2\left(y-7\right).

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{6y+18y-126}{2y\left(y-7\right)})$$

Combine like terms in 6y+18y-126.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{24y-126}{2y\left(y-7\right)})$$

Factor the expressions that are not already factored in \frac{24y-126}{2y\left(y-7\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{6\left(4y-21\right)}{2y\left(y-7\right)})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{3\left(4y-21\right)}{y\left(y-7\right)})$$

Use the distributive property to multiply 3 by 4y-21.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{12y-63}{y\left(y-7\right)})$$

Use the distributive property to multiply y by y-7.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{12y-63}{y^{2}-7y})$$

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.

$$\frac{\left(y^{2}-7y^{1}\right)\frac{\mathrm{d}}{\mathrm{d}y}(12y^{1}-63)-\left(12y^{1}-63\right)\frac{\mathrm{d}}{\mathrm{d}y}(y^{2}-7y^{1})}{\left(y^{2}-7y^{1}\right)^{2}}$$

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}.

$$\frac{\left(y^{2}-7y^{1}\right)\times 12y^{1-1}-\left(12y^{1}-63\right)\left(2y^{2-1}-7y^{1-1}\right)}{\left(y^{2}-7y^{1}\right)^{2}}$$

Simplify.

$$\frac{\left(y^{2}-7y^{1}\right)\times 12y^{0}-\left(12y^{1}-63\right)\left(2y^{1}-7y^{0}\right)}{\left(y^{2}-7y^{1}\right)^{2}}$$

Multiply y^{2}-7y^{1} times 12y^{0}.

$$\frac{y^{2}\times 12y^{0}-7y^{1}\times 12y^{0}-\left(12y^{1}-63\right)\left(2y^{1}-7y^{0}\right)}{\left(y^{2}-7y^{1}\right)^{2}}$$

Multiply 12y^{1}-63 times 2y^{1}-7y^{0}.

$$\frac{y^{2}\times 12y^{0}-7y^{1}\times 12y^{0}-\left(12y^{1}\times 2y^{1}+12y^{1}\left(-7\right)y^{0}-63\times 2y^{1}-63\left(-7\right)y^{0}\right)}{\left(y^{2}-7y^{1}\right)^{2}}$$

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

$$\frac{12y^{2}-7\times 12y^{1}-\left(12\times 2y^{1+1}+12\left(-7\right)y^{1}-63\times 2y^{1}-63\left(-7\right)y^{0}\right)}{\left(y^{2}-7y^{1}\right)^{2}}$$

Simplify.

$$\frac{12y^{2}-84y^{1}-\left(24y^{2}-84y^{1}-126y^{1}+441y^{0}\right)}{\left(y^{2}-7y^{1}\right)^{2}}$$

Combine like terms.

$$\frac{-12y^{2}+126y^{1}-441y^{0}}{\left(y^{2}-7y^{1}\right)^{2}}$$

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

$$\frac{-12y^{2}+126y-441y^{0}}{\left(y^{2}-7y\right)^{2}}$$

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

$$\frac{-12y^{2}+126y-441}{\left(y^{2}-7y\right)^{2}}$$