Solve x/x-1+1/x-9

$$\frac { x } { x - 1 } + \frac { 1 } { x - 9 }$$

Evaluate

$$\frac{x^{2}-8x-1}{\left(x-9\right)\left(x-1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x-9 is \left(x-9\right)\left(x-1\right). Multiply \frac{x}{x-1} times \frac{x-9}{x-9}. Multiply \frac{1}{x-9} times \frac{x-1}{x-1}.

$$\frac{x\left(x-9\right)}{\left(x-9\right)\left(x-1\right)}+\frac{x-1}{\left(x-9\right)\left(x-1\right)}$$

Since \frac{x\left(x-9\right)}{\left(x-9\right)\left(x-1\right)} and \frac{x-1}{\left(x-9\right)\left(x-1\right)} have the same denominator, add them by adding their numerators.

$$\frac{x\left(x-9\right)+x-1}{\left(x-9\right)\left(x-1\right)}$$

Do the multiplications in x\left(x-9\right)+x-1.

$$\frac{x^{2}-9x+x-1}{\left(x-9\right)\left(x-1\right)}$$

Combine like terms in x^{2}-9x+x-1.

$$\frac{x^{2}-8x-1}{\left(x-9\right)\left(x-1\right)}$$

Expand \left(x-9\right)\left(x-1\right).

$$\frac{x^{2}-8x-1}{x^{2}-10x+9}$$

Differentiate w.r.t. x

$$\frac{2\left(-x^{2}+10x-41\right)}{\left(\left(x-9\right)\left(x-1\right)\right)^{2}}$$