Solve (4x^2y^3)/(2x^3y)

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

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{2}y^{3}}{2^{1}x^{3}y^{1}}$$

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

$$\frac{4^{1}}{2^{1}}x^{2-3}y^{3-1}$$

Subtract 3 from 2.

$$\frac{4^{1}}{2^{1}}\times \frac{1}{x}y^{3-1}$$

Subtract 1 from 3.

$$\frac{4^{1}}{2^{1}}\times \frac{1}{x}y^{2}$$

Divide 4 by 2.

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

Differentiate w.r.t. x

$$-2\times \left(\frac{y}{x}\right)^{2}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2y^{2}\times \frac{1}{x})$$

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}.

$$-2y^{2}x^{-1-1}$$

Do the arithmetic.

$$\left(-2y^{2}\right)x^{-2}$$