Solve 30x^6y^5/20x^3y^2

$$\frac{ 30 { x }^{ 6 } { y }^{ 5 } }{ 20 { x }^{ 3 } { y }^{ 2 } }$$

Evaluate

$$\frac{3\left(xy\right)^{3}}{2}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{30^{1}x^{6}y^{5}}{20^{1}x^{3}y^{2}}$$

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

$$\frac{30^{1}}{20^{1}}x^{6-3}y^{5-2}$$

Subtract 3 from 6.

$$\frac{30^{1}}{20^{1}}x^{3}y^{5-2}$$

Subtract 2 from 5.

$$\frac{30^{1}}{20^{1}}x^{3}y^{3}$$

Reduce the fraction \frac{30}{20} to lowest terms by extracting and canceling out 10.

$$\frac{3}{2}x^{3}y^{3}$$

Differentiate w.r.t. x

$$\frac{9x^{2}y^{3}}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{30y^{5}}{20y^{2}}x^{6-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y^{3}}{2}x^{3})$$

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}.

$$3\times \frac{3y^{3}}{2}x^{3-1}$$

Do the arithmetic.

$$\frac{9y^{3}}{2}x^{2}$$