Solve d/dxleft(x^2-x/x^3-x^2-x+1right)

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

Evaluate

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

Steps Using Derivative Rule for Quotient

Factor the expressions that are not already factored in \frac{x^{2}-x}{x^{3}-x^{2}-x+1}.

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

Cancel out x-1 in both numerator and denominator.

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

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

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

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

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

Do the arithmetic.

$$\frac{\left(x^{2}-1\right)x^{0}-x^{1}\times 2x^{1}}{\left(x^{2}-1\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{2}x^{0}-x^{0}-x^{1}\times 2x^{1}}{\left(x^{2}-1\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{x^{2}-x^{0}-2x^{1+1}}{\left(x^{2}-1\right)^{2}}$$

Do the arithmetic.

$$\frac{x^{2}-x^{0}-2x^{2}}{\left(x^{2}-1\right)^{2}}$$

Combine like terms.

$$\frac{\left(1-2\right)x^{2}-x^{0}}{\left(x^{2}-1\right)^{2}}$$

Subtract 2 from 1.

$$\frac{-x^{2}-x^{0}}{\left(x^{2}-1\right)^{2}}$$

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

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

Differentiate w.r.t. x

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