Solve (x^frac{1{2}}y^frac{-2{3}})^-frac{5{4}}

$$( x ^ { \frac { 1 } { 2 } } y ^ { \frac { - 2 } { 3 } } ) ^ { - \frac { 5 } { 4 } }$$

Evaluate

$$\frac{\left(|y|\right)^{\frac{5}{6}}}{x^{\frac{5}{8}}}$$

Solution Steps

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

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

Expand \left(x^{\frac{1}{2}}y^{-\frac{2}{3}}\right)^{-\frac{5}{4}}.

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

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and -\frac{5}{4} to get -\frac{5}{8}.

$$x^{-\frac{5}{8}}\left(y^{-\frac{2}{3}}\right)^{-\frac{5}{4}}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{2}{3} and -\frac{5}{4} to get \frac{5}{6}.

$$x^{-\frac{5}{8}}y^{\frac{5}{6}}$$

Differentiate w.r.t. x

$$-\frac{5\left(|y|\right)^{\frac{5}{6}}}{8x^{\frac{13}{8}}}$$