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

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

Evaluate

$$\frac{5y}{2x^{6}}$$

Solution Steps

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

$$\frac{5x^{3}y^{5}}{3}\times \frac{3x^{-4}y^{-4}}{2x^{5}}$$

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

$$\frac{5x^{3}y^{5}}{3}\times \frac{3y^{-4}}{2x^{9}}$$

Multiply \frac{5x^{3}y^{5}}{3} times \frac{3y^{-4}}{2x^{9}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5x^{3}y^{5}\times 3y^{-4}}{3\times 2x^{9}}$$

Cancel out 3x^{3} in both numerator and denominator.

$$\frac{5y^{-4}y^{5}}{2x^{6}}$$

To multiply powers of the same base, add their exponents. Add -4 and 5 to get 1.

$$\frac{5y^{1}}{2x^{6}}$$

Calculate y to the power of 1 and get y.

$$\frac{5y}{2x^{6}}$$

Differentiate w.r.t. x

$$-\frac{15y}{x^{7}}$$