Solve 3x/x^2-4-1/x^2

$$\frac { 3 x } { x ^ { 2 } - 4 } - \frac { 1 } { x ^ { 2 } }$$

Evaluate

$$\frac{\left(x+1\right)\left(3x^{2}-4x+4\right)}{x^{2}\left(x^{2}-4\right)}$$

Short Solution Steps

Factor x^{2}-4.

$$\frac{3x}{\left(x-2\right)\left(x+2\right)}-\frac{1}{x^{2}}$$

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

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

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

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

Do the multiplications in 3xx^{2}-\left(x-2\right)\left(x+2\right).

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

Combine like terms in 3x^{3}-x^{2}-2x+2x+4.

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

Expand \left(x-2\right)\left(x+2\right)x^{2}.

$$\frac{3x^{3}-x^{2}+4}{x^{4}-4x^{2}}$$

Differentiate w.r.t. x

$$\frac{32-16x^{2}-12x^{3}+2x^{4}-3x^{5}}{x^{3}\left(x^{2}-4\right)^{2}}$$