Solve x^{frac{1/2}y^-frac{1{5}}}{x^frac{5{2}}y^1x^-frac{5{4}}}

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

Evaluate

$$\frac{1}{x^{\frac{3}{4}}y^{\frac{6}{5}}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{5}{2} and -\frac{5}{4} to get \frac{5}{4}.

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

Cancel out \sqrt{x} in both numerator and denominator.

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

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

$$\frac{1}{x^{\frac{3}{4}}y^{\frac{6}{5}}}$$

Differentiate w.r.t. x

$$-\frac{3}{4y^{\frac{6}{5}}x^{\frac{7}{4}}}$$