Solve 1/x-x^2-1/x

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

Evaluate

$$\frac{1}{1-x}$$

Short Solution Steps

Factor x-x^{2}.

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

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

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

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

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

Do the multiplications in 1-\left(-x+1\right).

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

Combine like terms in 1+x-1.

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

Cancel out x in both numerator and denominator.

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

Differentiate w.r.t. x

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

Steps Using Chain Rule

Factor x-x^{2}.

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

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

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

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

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

Do the multiplications in 1-\left(-x+1\right).

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

Combine like terms in 1+x-1.

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

Cancel out x in both numerator and denominator.

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

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

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

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}.

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

Simplify.

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

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

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

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

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

For any term t, t\times 1=t and 1t=t.

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