Solve (x^{frac{3/3}y^-frac{5{6}}}{x^frac{3{3}}y^frac{2{3}}})^frac{3{2}}

$$( \frac { x ^ { \frac { 3 } { 3 } } y ^ { - \frac { 5 } { 6 } } } { x ^ { \frac { 3 } { 3 } } y ^ { \frac { 2 } { 3 } } } ) ^ { \frac { 3 } { 2 } }$$

Differentiate w.r.t. y

$$-\frac{9}{4y^{\frac{13}{4}}}$$

Steps Using Chain Rule

Divide 3 by 3 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(\frac{x^{1}y^{-\frac{5}{6}}}{x^{\frac{3}{3}}y^{\frac{2}{3}}}\right)^{\frac{3}{2}})$$

Divide 3 by 3 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(\frac{x^{1}y^{-\frac{5}{6}}}{x^{1}y^{\frac{2}{3}}}\right)^{\frac{3}{2}})$$

Cancel out x^{1} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(\frac{y^{-\frac{5}{6}}}{y^{\frac{2}{3}}}\right)^{\frac{3}{2}})$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(\frac{1}{y^{\frac{3}{2}}}\right)^{\frac{3}{2}})$$

To raise \frac{1}{y^{\frac{3}{2}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1^{\frac{3}{2}}}{\left(y^{\frac{3}{2}}\right)^{\frac{3}{2}}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{2} and \frac{3}{2} to get \frac{9}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1^{\frac{3}{2}}}{y^{\frac{9}{4}}})$$

Calculate 1 to the power of \frac{3}{2} and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{\frac{9}{4}}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(y^{\frac{9}{4}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{\frac{9}{4}})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(y^{\frac{9}{4}}\right)^{-2}\times \frac{9}{4}y^{\frac{9}{4}-1}$$

Simplify.

$$-\frac{9}{4}y^{\frac{5}{4}}\left(y^{\frac{9}{4}}\right)^{-2}$$

Evaluate

$$\frac{1}{y^{\frac{9}{4}}}$$