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

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{x^{2}+3}{\left(x^{2}+1\right)\left(x^{2}+3\right)} and \frac{-\left(x^{2}+1\right)}{\left(x^{2}+1\right)\left(x^{2}+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x^{2}+3-\left(x^{2}+1\right).

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

Combine like terms in x^{2}+3-x^{2}-1.

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

Expand \left(x^{2}+1\right)\left(x^{2}+3\right).

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

Differentiate w.r.t. x

$$-\frac{8x\left(x^{2}+2\right)}{\left(\left(x^{2}+1\right)\left(x^{2}+3\right)\right)^{2}}$$