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Differentiate w.r.t. x
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\left(x^{\frac{1}{2}}y^{-\frac{2}{3}}\right)^{-\frac{3}{2}}
Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.
\left(x^{\frac{1}{2}}\right)^{-\frac{3}{2}}\left(y^{-\frac{2}{3}}\right)^{-\frac{3}{2}}
Expand \left(x^{\frac{1}{2}}y^{-\frac{2}{3}}\right)^{-\frac{3}{2}}.
x^{-\frac{3}{4}}\left(y^{-\frac{2}{3}}\right)^{-\frac{3}{2}}
To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and -\frac{3}{2} to get -\frac{3}{4}.
x^{-\frac{3}{4}}y^{1}
To raise a power to another power, multiply the exponents. Multiply -\frac{2}{3} and -\frac{3}{2} to get 1.
x^{-\frac{3}{4}}y
Calculate y to the power of 1 and get y.