Solve (a^2/a-1+1/1-a):(1+1/a)

$$( \frac { a ^ { 2 } } { a - 1 } + \frac { 1 } { 1 - a } ) : ( 1 + \frac { 1 } { a } )$$

Evaluate

$$a$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and 1-a is a-1. Multiply \frac{1}{1-a} times \frac{-1}{-1}.

$$\frac{\frac{a^{2}}{a-1}+\frac{-1}{a-1}}{1+\frac{1}{a}}$$

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

$$\frac{\frac{a^{2}-1}{a-1}}{1+\frac{1}{a}}$$

Factor the expressions that are not already factored in \frac{a^{2}-1}{a-1}.

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

Cancel out a-1 in both numerator and denominator.

$$\frac{a+1}{1+\frac{1}{a}}$$

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

$$\frac{a+1}{\frac{a}{a}+\frac{1}{a}}$$

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

$$\frac{a+1}{\frac{a+1}{a}}$$

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

$$\frac{\left(a+1\right)a}{a+1}$$

Cancel out a+1 in both numerator and denominator.

$$a$$

Differentiate w.r.t. a

$$1$$

Steps Using Definition of a Derivative

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and 1-a is a-1. Multiply \frac{1}{1-a} times \frac{-1}{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{2}}{a-1}+\frac{-1}{a-1}}{1+\frac{1}{a}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{2}-1}{a-1}}{1+\frac{1}{a}})$$

Factor the expressions that are not already factored in \frac{a^{2}-1}{a-1}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{\left(a-1\right)\left(a+1\right)}{a-1}}{1+\frac{1}{a}})$$

Cancel out a-1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a+1}{1+\frac{1}{a}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a+1}{\frac{a}{a}+\frac{1}{a}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a+1}{\frac{a+1}{a}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\left(a+1\right)a}{a+1})$$

Cancel out a+1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a)$$

The derivative of ax^{n} is nax^{n-1}.

$$a^{1-1}$$

Subtract 1 from 1.

$$a^{0}$$

For any term t except 0, t^{0}=1.

$$1$$