Solve frac{a/b^{2}*b/2a^2}{3ab/b^3*b^2/2a^2}

$$\frac { \frac { a } { b ^ { 2 } } \cdot \frac { b } { 2 a ^ { 2 } } } { \frac { 3 a b } { b ^ { 3 } } \cdot \frac { b ^ { 2 } } { 2 a ^ { 2 } } }$$

Evaluate

$$\frac{1}{3b}$$

Solution Steps

Multiply \frac{a}{b^{2}} times \frac{b}{2a^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out ab in both numerator and denominator.

$$\frac{\frac{1}{2ab}}{\frac{3ab}{b^{3}}\times \frac{b^{2}}{2a^{2}}}$$

Cancel out b in both numerator and denominator.

$$\frac{\frac{1}{2ab}}{\frac{3a}{b^{2}}\times \frac{b^{2}}{2a^{2}}}$$

Multiply \frac{3a}{b^{2}} times \frac{b^{2}}{2a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{1}{2ab}}{\frac{3ab^{2}}{b^{2}\times 2a^{2}}}$$

Cancel out ab^{2} in both numerator and denominator.

$$\frac{\frac{1}{2ab}}{\frac{3}{2a}}$$

Divide \frac{1}{2ab} by \frac{3}{2a} by multiplying \frac{1}{2ab} by the reciprocal of \frac{3}{2a}.

$$\frac{2a}{2ab\times 3}$$

Cancel out 2a in both numerator and denominator.

$$\frac{1}{3b}$$

Differentiate w.r.t. b

$$-\frac{1}{3b^{2}}$$