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

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

Divide \frac{1}{x_{1}} by \frac{x_{1}x_{2}+1}{x_{1}x_{2}} by multiplying \frac{1}{x_{1}} by the reciprocal of \frac{x_{1}x_{2}+1}{x_{1}x_{2}}.

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

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

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

Differentiate w.r.t. x_1

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