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

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

Evaluate

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

Short Solution Steps

Factor x^{2}-4. Factor x^{2}-4x+4.

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

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 \left(x-2\right)^{2} is \left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{1}{\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{1}{\left(x-2\right)^{2}} times \frac{x+2}{x+2}.

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

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

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

Combine like terms in x-2+x+2.

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

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

$$\frac{2x}{x^{3}-2x^{2}-4x+8}$$

Differentiate w.r.t. x

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