Solve a/a+a{a+frac{b/a}}

$$\frac { a } { a + \frac { a } { a + \frac { b } { a } } }$$

Evaluate

$$\frac{a^{2}+b}{a^{2}+a+b}$$

Short Solution Steps

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

$$\frac{a}{a+\frac{a}{\frac{aa}{a}+\frac{b}{a}}}$$

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

$$\frac{a}{a+\frac{a}{\frac{aa+b}{a}}}$$

Do the multiplications in aa+b.

$$\frac{a}{a+\frac{a}{\frac{a^{2}+b}{a}}}$$

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

$$\frac{a}{a+\frac{aa}{a^{2}+b}}$$

Multiply a and a to get a^{2}.

$$\frac{a}{a+\frac{a^{2}}{a^{2}+b}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a^{2}+b}{a^{2}+b}.

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

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

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

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

$$\frac{a}{\frac{a^{3}+ab+a^{2}}{a^{2}+b}}$$

Divide a by \frac{a^{3}+ab+a^{2}}{a^{2}+b} by multiplying a by the reciprocal of \frac{a^{3}+ab+a^{2}}{a^{2}+b}.

$$\frac{a\left(a^{2}+b\right)}{a^{3}+ab+a^{2}}$$

Factor the expressions that are not already factored.

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

Cancel out a in both numerator and denominator.

$$\frac{a^{2}+b}{a^{2}+a+b}$$

Differentiate w.r.t. a

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