Solve 1/x^2+1/frac{1{x^2}+x}-1/x

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x^{2}}{x^{2}}.

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

Since \frac{1}{x^{2}} and \frac{xx^{2}}{x^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 1+xx^{2}.

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

Divide 1 by \frac{1+x^{3}}{x^{2}} by multiplying 1 by the reciprocal of \frac{1+x^{3}}{x^{2}}.

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

Factor 1+x^{3}.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{x^{3}+1-x}{x^{5}+x^{2}}$$

Differentiate w.r.t. x

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