Solve 5x/x^2-x-6-18/x^2-9

$$\frac { 5 x } { x ^ { 2 } - x - 6 } - \frac { 18 } { x ^ { 2 } - 9 }$$

Evaluate

$$\frac{5x+12}{\left(x+2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}-x-6. Factor x^{2}-9.

$$\frac{5x}{\left(x-3\right)\left(x+2\right)}-\frac{18}{\left(x-3\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+2\right) and \left(x-3\right)\left(x+3\right) is \left(x-3\right)\left(x+2\right)\left(x+3\right). Multiply \frac{5x}{\left(x-3\right)\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{18}{\left(x-3\right)\left(x+3\right)} times \frac{x+2}{x+2}.

$$\frac{5x\left(x+3\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}-\frac{18\left(x+2\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}$$

Since \frac{5x\left(x+3\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)} and \frac{18\left(x+2\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x\left(x+3\right)-18\left(x+2\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}$$

Do the multiplications in 5x\left(x+3\right)-18\left(x+2\right).

$$\frac{5x^{2}+15x-18x-36}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}$$

Combine like terms in 5x^{2}+15x-18x-36.

$$\frac{5x^{2}-3x-36}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{5x^{2}-3x-36}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}.

$$\frac{\left(x-3\right)\left(5x+12\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{5x+12}{\left(x+2\right)\left(x+3\right)}$$

Expand \left(x+2\right)\left(x+3\right).

$$\frac{5x+12}{x^{2}+5x+6}$$

Differentiate w.r.t. x

$$-\frac{5x^{2}+24x+30}{\left(\left(x+2\right)\left(x+3\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Factor x^{2}-x-6. Factor x^{2}-9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x}{\left(x-3\right)\left(x+2\right)}-\frac{18}{\left(x-3\right)\left(x+3\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+2\right) and \left(x-3\right)\left(x+3\right) is \left(x-3\right)\left(x+2\right)\left(x+3\right). Multiply \frac{5x}{\left(x-3\right)\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{18}{\left(x-3\right)\left(x+3\right)} times \frac{x+2}{x+2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x\left(x+3\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}-\frac{18\left(x+2\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)})$$

Since \frac{5x\left(x+3\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)} and \frac{18\left(x+2\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x\left(x+3\right)-18\left(x+2\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)})$$

Do the multiplications in 5x\left(x+3\right)-18\left(x+2\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x^{2}+15x-18x-36}{\left(x-3\right)\left(x+2\right)\left(x+3\right)})$$

Combine like terms in 5x^{2}+15x-18x-36.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x^{2}-3x-36}{\left(x-3\right)\left(x+2\right)\left(x+3\right)})$$

Factor the expressions that are not already factored in \frac{5x^{2}-3x-36}{\left(x-3\right)\left(x+2\right)\left(x+3\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-3\right)\left(5x+12\right)}{\left(x-3\right)\left(x+2\right)\left(x+3\right)})$$

Cancel out x-3 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x+12}{\left(x+2\right)\left(x+3\right)})$$

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x+12}{x^{2}+5x+6})$$

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.

$$\frac{\left(x^{2}+5x^{1}+6\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1}+12)-\left(5x^{1}+12\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+5x^{1}+6)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

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}.

$$\frac{\left(x^{2}+5x^{1}+6\right)\times 5x^{1-1}-\left(5x^{1}+12\right)\left(2x^{2-1}+5x^{1-1}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Simplify.

$$\frac{\left(x^{2}+5x^{1}+6\right)\times 5x^{0}-\left(5x^{1}+12\right)\left(2x^{1}+5x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Multiply x^{2}+5x^{1}+6 times 5x^{0}.

$$\frac{x^{2}\times 5x^{0}+5x^{1}\times 5x^{0}+6\times 5x^{0}-\left(5x^{1}+12\right)\left(2x^{1}+5x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Multiply 5x^{1}+12 times 2x^{1}+5x^{0}.

$$\frac{x^{2}\times 5x^{0}+5x^{1}\times 5x^{0}+6\times 5x^{0}-\left(5x^{1}\times 2x^{1}+5x^{1}\times 5x^{0}+12\times 2x^{1}+12\times 5x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

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

$$\frac{5x^{2}+5\times 5x^{1}+6\times 5x^{0}-\left(5\times 2x^{1+1}+5\times 5x^{1}+12\times 2x^{1}+12\times 5x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Simplify.

$$\frac{5x^{2}+25x^{1}+30x^{0}-\left(10x^{2}+25x^{1}+24x^{1}+60x^{0}\right)}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

Combine like terms.

$$\frac{-5x^{2}-24x^{1}-30x^{0}}{\left(x^{2}+5x^{1}+6\right)^{2}}$$

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

$$\frac{-5x^{2}-24x-30x^{0}}{\left(x^{2}+5x+6\right)^{2}}$$

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

$$\frac{-5x^{2}-24x-30}{\left(x^{2}+5x+6\right)^{2}}$$