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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{4}-x+1-x^{4}-x-1.

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

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

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

Differentiate w.r.t. x

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