Solve a^5/a-1-a^2/a+1-1/a-1+1/a+1

$$\frac { a ^ { 5 } } { a - 1 } - \frac { a ^ { 2 } } { a + 1 } - \frac { 1 } { a - 1 } + \frac { 1 } { a + 1 }$$

Evaluate

$$a^{4}+a^{3}+a^{2}+2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and a+1 is \left(a-1\right)\left(a+1\right). Multiply \frac{a^{5}}{a-1} times \frac{a+1}{a+1}. Multiply \frac{a^{2}}{a+1} times \frac{a-1}{a-1}.

$$\frac{a^{5}\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}-\frac{a^{2}\left(a-1\right)}{\left(a-1\right)\left(a+1\right)}-\frac{1}{a-1}+\frac{1}{a+1}$$

Since \frac{a^{5}\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} and \frac{a^{2}\left(a-1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a^{5}\left(a+1\right)-a^{2}\left(a-1\right)}{\left(a-1\right)\left(a+1\right)}-\frac{1}{a-1}+\frac{1}{a+1}$$

Do the multiplications in a^{5}\left(a+1\right)-a^{2}\left(a-1\right).

$$\frac{a^{6}+a^{5}-a^{3}+a^{2}}{\left(a-1\right)\left(a+1\right)}-\frac{1}{a-1}+\frac{1}{a+1}$$

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

$$\frac{a^{6}+a^{5}-a^{3}+a^{2}}{\left(a-1\right)\left(a+1\right)}-\frac{a+1}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a+1}$$

Since \frac{a^{6}+a^{5}-a^{3}+a^{2}}{\left(a-1\right)\left(a+1\right)} and \frac{a+1}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a^{6}+a^{5}-a^{3}+a^{2}-\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a+1}$$

Do the multiplications in a^{6}+a^{5}-a^{3}+a^{2}-\left(a+1\right).

$$\frac{a^{6}+a^{5}-a^{3}+a^{2}-a-1}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a+1}$$

Factor the expressions that are not already factored in \frac{a^{6}+a^{5}-a^{3}+a^{2}-a-1}{\left(a-1\right)\left(a+1\right)}.

$$\frac{\left(a-1\right)\left(a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1\right)}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a+1}$$

Cancel out a-1 in both numerator and denominator.

$$\frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1}{a+1}+\frac{1}{a+1}$$

Since \frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1}{a+1} and \frac{1}{a+1} have the same denominator, add them by adding their numerators.

$$\frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1+1}{a+1}$$

Combine like terms in a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1+1.

$$\frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+2}{a+1}$$

Factor the expressions that are not already factored in \frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+2}{a+1}.

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

Cancel out a+1 in both numerator and denominator.

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

Expand the expression.

$$a^{4}+a^{3}+a^{2}+2$$

Differentiate w.r.t. a

$$a\left(4a^{2}+3a+2\right)$$

Steps Using Derivative Rule for Sum

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and a+1 is \left(a-1\right)\left(a+1\right). Multiply \frac{a^{5}}{a-1} times \frac{a+1}{a+1}. Multiply \frac{a^{2}}{a+1} times \frac{a-1}{a-1}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{5}\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}-\frac{a^{2}\left(a-1\right)}{\left(a-1\right)\left(a+1\right)}-\frac{1}{a-1}+\frac{1}{a+1})$$

Since \frac{a^{5}\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} and \frac{a^{2}\left(a-1\right)}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{5}\left(a+1\right)-a^{2}\left(a-1\right)}{\left(a-1\right)\left(a+1\right)}-\frac{1}{a-1}+\frac{1}{a+1})$$

Do the multiplications in a^{5}\left(a+1\right)-a^{2}\left(a-1\right).

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{6}+a^{5}-a^{3}+a^{2}}{\left(a-1\right)\left(a+1\right)}-\frac{1}{a-1}+\frac{1}{a+1})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{6}+a^{5}-a^{3}+a^{2}}{\left(a-1\right)\left(a+1\right)}-\frac{a+1}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a+1})$$

Since \frac{a^{6}+a^{5}-a^{3}+a^{2}}{\left(a-1\right)\left(a+1\right)} and \frac{a+1}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{6}+a^{5}-a^{3}+a^{2}-\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a+1})$$

Do the multiplications in a^{6}+a^{5}-a^{3}+a^{2}-\left(a+1\right).

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{6}+a^{5}-a^{3}+a^{2}-a-1}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a+1})$$

Factor the expressions that are not already factored in \frac{a^{6}+a^{5}-a^{3}+a^{2}-a-1}{\left(a-1\right)\left(a+1\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\left(a-1\right)\left(a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1\right)}{\left(a-1\right)\left(a+1\right)}+\frac{1}{a+1})$$

Cancel out a-1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1}{a+1}+\frac{1}{a+1})$$

Since \frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1}{a+1} and \frac{1}{a+1} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1+1}{a+1})$$

Combine like terms in a^{5}+2a^{4}+2a^{3}+a^{2}+2a+1+1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+2}{a+1})$$

Factor the expressions that are not already factored in \frac{a^{5}+2a^{4}+2a^{3}+a^{2}+2a+2}{a+1}.

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

Cancel out a+1 in both numerator and denominator.

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

Expand the expression.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{4}+a^{3}+a^{2}+2)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$4a^{4-1}+3a^{3-1}+2a^{2-1}$$

Subtract 1 from 4.

$$4a^{3}+3a^{3-1}+2a^{2-1}$$

Subtract 1 from 3.

$$4a^{3}+3a^{2}+2a^{2-1}$$

Subtract 1 from 2.

$$4a^{3}+3a^{2}+2a^{1}$$

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

$$4a^{3}+3a^{2}+2a$$