Solve x/x^2-3x-4-1/x-4

$$\frac { x } { x ^ { 2 } - 3 x - 4 } - \frac { 1 } { x - 4 }$$

Evaluate

$$-\frac{1}{\left(x-4\right)\left(x+1\right)}$$

Short Solution Steps

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

$$\frac{x}{\left(x-4\right)\left(x+1\right)}-\frac{1}{x-4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x+1\right) and x-4 is \left(x-4\right)\left(x+1\right). Multiply \frac{1}{x-4} times \frac{x+1}{x+1}.

$$\frac{x}{\left(x-4\right)\left(x+1\right)}-\frac{x+1}{\left(x-4\right)\left(x+1\right)}$$

Since \frac{x}{\left(x-4\right)\left(x+1\right)} and \frac{x+1}{\left(x-4\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-\left(x+1\right)}{\left(x-4\right)\left(x+1\right)}$$

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

$$\frac{x-x-1}{\left(x-4\right)\left(x+1\right)}$$

Combine like terms in x-x-1.

$$\frac{-1}{\left(x-4\right)\left(x+1\right)}$$

Expand \left(x-4\right)\left(x+1\right).

$$\frac{-1}{x^{2}-3x-4}$$

Differentiate w.r.t. x

$$\frac{2x-3}{\left(\left(x-4\right)\left(x+1\right)\right)^{2}}$$