Solve x/x^2-9-3/x^2+3x

$$\frac { x } { x ^ { 2 } - 9 } - \frac { 3 } { x ^ { 2 } + 3 x }$$

Evaluate

$$\frac{x^{2}-3x+9}{x\left(x^{2}-9\right)}$$

Short Solution Steps

Factor x^{2}-9. Factor x^{2}+3x.

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

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

$$\frac{xx}{x\left(x-3\right)\left(x+3\right)}-\frac{3\left(x-3\right)}{x\left(x-3\right)\left(x+3\right)}$$

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

$$\frac{xx-3\left(x-3\right)}{x\left(x-3\right)\left(x+3\right)}$$

Do the multiplications in xx-3\left(x-3\right).

$$\frac{x^{2}-3x+9}{x\left(x-3\right)\left(x+3\right)}$$

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

$$\frac{x^{2}-3x+9}{x^{3}-9x}$$

Differentiate w.r.t. x

$$\frac{81-36x^{2}+6x^{3}-x^{4}}{\left(x\left(x^{2}-9\right)\right)^{2}}$$