Solve x/x^2+17x+72-1/x^2+15x+56

$$\frac { x } { x ^ { 2 } + 17 x + 72 } - \frac { 1 } { x ^ { 2 } + 15 x + 56 }$$

Evaluate

$$\frac{x^{2}+6x-9}{\left(x+7\right)\left(x+8\right)\left(x+9\right)}$$

Short Solution Steps

Factor x^{2}+17x+72. Factor x^{2}+15x+56.

$$\frac{x}{\left(x+8\right)\left(x+9\right)}-\frac{1}{\left(x+7\right)\left(x+8\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+8\right)\left(x+9\right) and \left(x+7\right)\left(x+8\right) is \left(x+7\right)\left(x+8\right)\left(x+9\right). Multiply \frac{x}{\left(x+8\right)\left(x+9\right)} times \frac{x+7}{x+7}. Multiply \frac{1}{\left(x+7\right)\left(x+8\right)} times \frac{x+9}{x+9}.

$$\frac{x\left(x+7\right)}{\left(x+7\right)\left(x+8\right)\left(x+9\right)}-\frac{x+9}{\left(x+7\right)\left(x+8\right)\left(x+9\right)}$$

Since \frac{x\left(x+7\right)}{\left(x+7\right)\left(x+8\right)\left(x+9\right)} and \frac{x+9}{\left(x+7\right)\left(x+8\right)\left(x+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+7\right)-\left(x+9\right)}{\left(x+7\right)\left(x+8\right)\left(x+9\right)}$$

Do the multiplications in x\left(x+7\right)-\left(x+9\right).

$$\frac{x^{2}+7x-x-9}{\left(x+7\right)\left(x+8\right)\left(x+9\right)}$$

Combine like terms in x^{2}+7x-x-9.

$$\frac{x^{2}+6x-9}{\left(x+7\right)\left(x+8\right)\left(x+9\right)}$$

Expand \left(x+7\right)\left(x+8\right)\left(x+9\right).

$$\frac{x^{2}+6x-9}{x^{3}+24x^{2}+191x+504}$$

Differentiate w.r.t. x

$$\frac{4743+1440x+74x^{2}-12x^{3}-x^{4}}{\left(\left(x+7\right)\left(x+8\right)\left(x+9\right)\right)^{2}}$$