Solve 2a^-1*b^-1/b^-1+2a^-1

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{2\times \frac{1}{a}\times \frac{1}{b}}{\left(2\times \frac{1}{a}b+1\right)\times \frac{1}{b}}$$

Cancel out \frac{1}{b} in both numerator and denominator.

$$\frac{2\times \frac{1}{a}}{2\times \frac{1}{a}b+1}$$

Express 2\times \frac{1}{a} as a single fraction.

$$\frac{\frac{2}{a}}{2\times \frac{1}{a}b+1}$$

Express 2\times \frac{1}{a} as a single fraction.

$$\frac{\frac{2}{a}}{\frac{2}{a}b+1}$$

Express \frac{2}{a}b as a single fraction.

$$\frac{\frac{2}{a}}{\frac{2b}{a}+1}$$

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

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

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

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

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

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

Cancel out a in both numerator and denominator.

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

Differentiate w.r.t. b

$$-\frac{4}{\left(a+2b\right)^{2}}$$