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

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

Evaluate

$$\frac{2\left(-x^{7}-x^{6}-x^{5}+3x^{4}-x^{3}-x^{2}-x-1\right)}{1-x^{8}}$$

Short Solution Steps

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

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

Since \frac{-x+1}{\left(x+1\right)\left(-x+1\right)} and \frac{x+1}{\left(x+1\right)\left(-x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -x+1-\left(x+1\right).

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

Combine like terms in -x+1-x-1.

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right) and 1+x^{4} is \left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right). Multiply \frac{-2x^{3}-2x-2x^{2}+2}{\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)} times \frac{x^{4}+1}{x^{4}+1}. Multiply \frac{4}{1+x^{4}} times \frac{\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)}{\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)}.

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

Since \frac{\left(-2x^{3}-2x-2x^{2}+2\right)\left(x^{4}+1\right)}{\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} and \frac{4\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)}{\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(-2x^{3}-2x-2x^{2}+2\right)\left(x^{4}+1\right)-4\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right).

$$\frac{-2x^{7}-2x^{3}-2x^{5}-2x-2x^{6}-2x^{2}+2x^{4}+2+4x^{4}+4x^{2}-4x^{3}-4x+4x^{3}+4x-4x^{2}-4}{\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}$$

Combine like terms in -2x^{7}-2x^{3}-2x^{5}-2x-2x^{6}-2x^{2}+2x^{4}+2+4x^{4}+4x^{2}-4x^{3}-4x+4x^{3}+4x-4x^{2}-4.

$$\frac{-2x^{7}-2x^{3}-2x^{5}-2x^{6}-2x-2x^{2}+6x^{4}-2}{\left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}$$

Expand \left(x+1\right)\left(-x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right).

$$\frac{-2x^{7}-2x^{3}-2x^{5}-2x^{6}-2x-2x^{2}+6x^{4}-2}{-x^{8}+1}$$

Differentiate w.r.t. x

$$-\frac{2\left(x^{14}+2x^{13}+3x^{12}-12x^{11}+5x^{10}+6x^{9}+7x^{8}+8x^{7}+7x^{6}+6x^{5}+5x^{4}-12x^{3}+3x^{2}+2x+1\right)}{\left(x^{8}-1\right)^{2}}$$