Solve a1/a+1+frac{2{a-1}}div(1/2-1/a+frac{1{a}})

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

Evaluate

$$\frac{2a_{1}}{a-1}$$

Short Solution Steps

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

$$\frac{\frac{a_{1}}{\frac{\left(a+1\right)\left(a-1\right)}{a-1}+\frac{2}{a-1}}}{\frac{1}{2}-\frac{1}{a+\frac{1}{a}}}$$

Since \frac{\left(a+1\right)\left(a-1\right)}{a-1} and \frac{2}{a-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{a_{1}}{\frac{\left(a+1\right)\left(a-1\right)+2}{a-1}}}{\frac{1}{2}-\frac{1}{a+\frac{1}{a}}}$$

Do the multiplications in \left(a+1\right)\left(a-1\right)+2.

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

Combine like terms in a^{2}-a+a-1+2.

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

Divide a_{1} by \frac{a^{2}+1}{a-1} by multiplying a_{1} by the reciprocal of \frac{a^{2}+1}{a-1}.

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

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

$$\frac{\frac{a_{1}\left(a-1\right)}{a^{2}+1}}{\frac{1}{2}-\frac{1}{\frac{aa}{a}+\frac{1}{a}}}$$

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

$$\frac{\frac{a_{1}\left(a-1\right)}{a^{2}+1}}{\frac{1}{2}-\frac{1}{\frac{aa+1}{a}}}$$

Do the multiplications in aa+1.

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

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

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

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

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

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

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

Divide \frac{a_{1}\left(a-1\right)}{a^{2}+1} by \frac{a^{2}+1-2a}{2\left(a^{2}+1\right)} by multiplying \frac{a_{1}\left(a-1\right)}{a^{2}+1} by the reciprocal of \frac{a^{2}+1-2a}{2\left(a^{2}+1\right)}.

$$\frac{a_{1}\left(a-1\right)\times 2\left(a^{2}+1\right)}{\left(a^{2}+1\right)\left(a^{2}+1-2a\right)}$$

Cancel out a^{2}+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

$$\frac{2a_{1}\left(a-1\right)}{\left(a-1\right)^{2}}$$

Cancel out a-1 in both numerator and denominator.

$$\frac{2a_{1}}{a-1}$$

Differentiate w.r.t. a_1

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