Solve 1+(frac{x_{1}x_{2}}{1+x_{1}x_{2}}*1/x_{1})

$$1 + ( \frac { x _ { 1 } x _ { 2 } } { 1 + x _ { 1 } x _ { 2 } } \times \frac { 1 } { x _ { 1 } } )$$

Evaluate

$$\frac{x_{1}x_{2}+x_{2}+1}{x_{1}x_{2}+1}$$

Short Solution Steps

Multiply \frac{x_{1}x_{2}}{1+x_{1}x_{2}} times \frac{1}{x_{1}} by multiplying numerator times numerator and denominator times denominator.

$$1+\frac{x_{1}x_{2}}{\left(1+x_{1}x_{2}\right)x_{1}}$$

Cancel out x_{1} in both numerator and denominator.

$$1+\frac{x_{2}}{x_{1}x_{2}+1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x_{1}x_{2}+1}{x_{1}x_{2}+1}.

$$\frac{x_{1}x_{2}+1}{x_{1}x_{2}+1}+\frac{x_{2}}{x_{1}x_{2}+1}$$

Since \frac{x_{1}x_{2}+1}{x_{1}x_{2}+1} and \frac{x_{2}}{x_{1}x_{2}+1} have the same denominator, add them by adding their numerators.

$$\frac{x_{1}x_{2}+1+x_{2}}{x_{1}x_{2}+1}$$

Differentiate w.r.t. x_1

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