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

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

Evaluate

$$x$$

Short Solution Steps

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

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

Factor the expressions that are not already factored.

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

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

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

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

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

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

The derivative of ax^{n} is nax^{n-1}.

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

For any term t except 0, t^{0}=1.

$$1$$