Simplify f(x)=(1/x-1/(x+1))/(1/x+1/(x+1))

$f (x) = \frac{\frac{1}{x} - \frac{1}{x + 1}}{\frac{1}{x} + \frac{1}{x + 1}}$

Multiply the numerator and denominator of the complex fraction by $x (x + 1)$ .

Multiply $\frac{\frac{1}{x} - \frac{1}{x + 1}}{\frac{1}{x} + \frac{1}{x + 1}}$ by $\frac{x (x + 1)}{x (x + 1)}$ .

$f (x) = \frac{x (x + 1)}{x (x + 1)} \cdot \frac{\frac{1}{x} - \frac{1}{x + 1}}{\frac{1}{x} + \frac{1}{x + 1}}$

Combine.

$f (x) = \frac{x (x + 1) (\frac{1}{x} - \frac{1}{x + 1})}{x (x + 1) (\frac{1}{x} + \frac{1}{x + 1})}$

Apply the distributive property.

$f (x) = \frac{x (x + 1) (\frac{1}{x}) + x (x + 1) (- \frac{1}{x + 1})}{x (x + 1) (\frac{1}{x}) + x (x + 1) (\frac{1}{x + 1})}$

Simplify by cancelling.

Cancel the common factor of $x$ .

Cancel the common factor.

$f (x) = \frac{x (x + 1) (\frac{1}{x}) + x (x + 1) (- \frac{1}{x + 1})}{x (x + 1) (\frac{1}{x}) + x (x + 1) (\frac{1}{x + 1})}$

Rewrite the expression.

$f (x) = \frac{x + 1 + x (x + 1) (- \frac{1}{x + 1})}{x (x + 1) (\frac{1}{x}) + x (x + 1) (\frac{1}{x + 1})}$

Cancel the common factor of $x + 1$ .

Move the leading negative in $- \frac{1}{x + 1}$ into the numerator.

$f (x) = \frac{x + 1 + x (x + 1) (\frac{- 1}{x + 1})}{x (x + 1) (\frac{1}{x}) + x (x + 1) (\frac{1}{x + 1})}$

Factor $x + 1$ out of $x (x + 1)$ .

$f (x) = \frac{x + 1 + (x + 1) x (\frac{- 1}{x + 1})}{x (x + 1) (\frac{1}{x}) + x (x + 1) (\frac{1}{x + 1})}$

Cancel the common factor.

$f (x) = \frac{x + 1 + (x + 1) x (\frac{- 1}{x + 1})}{x (x + 1) (\frac{1}{x}) + x (x + 1) (\frac{1}{x + 1})}$

Rewrite the expression.

$f (x) = \frac{x + 1 + x \cdot - 1}{x (x + 1) (\frac{1}{x}) + x (x + 1) (\frac{1}{x + 1})}$

Cancel the common factor of $x$ .

Cancel the common factor.

$f (x) = \frac{x + 1 + x \cdot - 1}{x (x + 1) (\frac{1}{x}) + x (x + 1) (\frac{1}{x + 1})}$

Rewrite the expression.

$f (x) = \frac{x + 1 + x \cdot - 1}{x + 1 + x (x + 1) (\frac{1}{x + 1})}$

Cancel the common factor of $x + 1$ .

Factor $x + 1$ out of $x (x + 1)$ .

$f (x) = \frac{x + 1 + x \cdot - 1}{x + 1 + (x + 1) x (\frac{1}{x + 1})}$

Cancel the common factor.

$f (x) = \frac{x + 1 + x \cdot - 1}{x + 1 + (x + 1) x (\frac{1}{x + 1})}$

Rewrite the expression.

$f (x) = \frac{x + 1 + x \cdot - 1}{x + 1 + x}$

Simplify the numerator.

Move $- 1$ to the left of $x$ .

$f (x) = \frac{x + 1 - 1 \cdot x}{x + 1 + x}$

Rewrite $- 1 x$ as $- x$ .

$f (x) = \frac{x + 1 - x}{x + 1 + x}$

Subtract $x$ from $x$ .

$f (x) = \frac{0 + 1}{x + 1 + x}$

Add $0$ and $1$ .

$f (x) = \frac{1}{x + 1 + x}$

Add $x$ and $x$ .

$f (x) = \frac{1}{2 x + 1}$

Simplify f(x)=(1/x-1/(x+1))/(1/x+1/(x+1))

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