Solve 4x^2y^3/16x^3y

$$\frac { 4 x ^ { 2 } y ^ { 3 } } { 16 x ^ { 3 } y }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{2}y^{3}}{16^{1}x^{3}y^{1}}$$

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

$$\frac{4^{1}}{16^{1}}x^{2-3}y^{3-1}$$

Subtract 3 from 2.

$$\frac{4^{1}}{16^{1}}\times \frac{1}{x}y^{3-1}$$

Subtract 1 from 3.

$$\frac{4^{1}}{16^{1}}\times \frac{1}{x}y^{2}$$

Reduce the fraction \frac{4}{16} to lowest terms by extracting and canceling out 4.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y^{3}}{16y}x^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{2}}{4}\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

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