Solve 1/x+1-2x/x^2+1-4x/x^4+1-8x/x^8-1

$$\frac { 1 } { x + 1 } - \frac { 2 x } { x ^ { 2 } + 1 } - \frac { 4 x } { x ^ { 4 } + 1 } - \frac { 8 x } { x ^ { 8 } - 1 }$$

Evaluate

$$-\frac{1}{x-1}$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Since \frac{\left(-x^{2}+1-2x\right)\left(x^{4}+1\right)}{\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} and \frac{4x\left(x+1\right)\left(x^{2}+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.

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

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

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

Combine like terms in -x^{6}-x^{2}+x^{4}+1-2x^{5}-2x-4x^{4}-4x^{2}-4x^{3}-4x.

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

Factor x^{8}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right) and \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right) is \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right). Multiply \frac{-6x-x^{6}-5x^{2}-3x^{4}+1-2x^{5}-4x^{3}}{\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} times \frac{x-1}{x-1}.

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

Since \frac{\left(-6x-x^{6}-5x^{2}-3x^{4}+1-2x^{5}-4x^{3}\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)} and \frac{8x}{\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.

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

Do the multiplications in \left(-6x-x^{6}-5x^{2}-3x^{4}+1-2x^{5}-4x^{3}\right)\left(x-1\right)-8x.

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

Combine like terms in -6x^{2}+6x-x^{7}+x^{6}-5x^{3}+5x^{2}-3x^{5}+3x^{4}+x-1-2x^{6}+2x^{5}-4x^{4}+4x^{3}-8x.

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

Factor the expressions that are not already factored in \frac{-x^{2}-x^{6}-x-x^{7}-x^{3}-x^{5}-x^{4}-1}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)}.

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

Extract the negative sign in -1-x.

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

Cancel out \left(x+1\right)\left(x^{2}+1\right)\left(x^{4}+1\right) in both numerator and denominator.

$$\frac{-1}{x-1}$$

Differentiate w.r.t. x

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