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

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

Evaluate

$$a$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out a-1 in both numerator and denominator.

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

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

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

Cancel out a+1 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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. Multiply a times \frac{a+1}{a+1}.

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

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

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

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

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

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

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

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

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

Cancel out a-1 in both numerator and denominator.

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

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

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

Cancel out a+1 in both numerator and denominator.

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

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

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

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.

$$1$$