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

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

Evaluate

$$-\frac{6}{\left(x-2\right)\left(x-1\right)\left(x+4\right)}$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+4\right) and \left(x-2\right)\left(x-1\right) is \left(x-2\right)\left(x-1\right)\left(x+4\right). Multiply \frac{1}{\left(x-1\right)\left(x+4\right)} times \frac{x-2}{x-2}. Multiply \frac{1}{\left(x-2\right)\left(x-1\right)} times \frac{x+4}{x+4}.

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

Since \frac{x-2}{\left(x-2\right)\left(x-1\right)\left(x+4\right)} and \frac{x+4}{\left(x-2\right)\left(x-1\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x-2-\left(x+4\right).

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

Combine like terms in x-2-x-4.

$$\frac{-6}{\left(x-2\right)\left(x-1\right)\left(x+4\right)}$$

Expand \left(x-2\right)\left(x-1\right)\left(x+4\right).

$$\frac{-6}{x^{3}+x^{2}-10x+8}$$

Differentiate w.r.t. x

$$\frac{2\left(\left(3x+1\right)^{2}-31\right)}{\left(\left(x-2\right)\left(x-1\right)\left(x+4\right)\right)^{2}}$$