Solve 1/x^{2-4x+4}-1/x^{3-3x^{2}+4})^{3}*[frac{(x-2)^{4}(x+1)^{2}}{x^2}]^2*(x^2-4x+4/x)^-1

$$\frac { 1 } { x ^ { 2 } - 4 x + 4 } - \frac { 1 } { x ^ { 3 } - 3 x ^ { 2 } + 4 } ) ^ { 3 } \cdot [ \frac { ( x - 2 ) ^ { 4 } ( x + 1 ) ^ { 2 } } { x ^ { 2 } } ] ^ { 2 } \cdot ( \frac { x ^ { 2 } - 4 x + 4 } { x } ) ^ { - 1 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-4x+4. Factor x^{3}-3x^{2}+4.

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

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

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

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

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

Combine like terms in x+1-1.

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

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

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

Differentiate w.r.t. x

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