Solve 40xy^5/16x^2y^3

$$\frac { 40 x y ^ { 5 } } { 16 x ^ { 2 } y ^ { 3 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{40^{1}x^{1}y^{5}}{16^{1}x^{2}y^{3}}$$

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

$$\frac{40^{1}}{16^{1}}x^{1-2}y^{5-3}$$

Subtract 2 from 1.

$$\frac{40^{1}}{16^{1}}\times \frac{1}{x}y^{5-3}$$

Subtract 3 from 5.

$$\frac{40^{1}}{16^{1}}\times \frac{1}{x}y^{2}$$

Reduce the fraction \frac{40}{16} to lowest terms by extracting and canceling out 8.

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

Differentiate w.r.t. x

$$-\frac{5\times \left(\frac{y}{x}\right)^{2}}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{40y^{5}}{16y^{3}}x^{1-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5y^{2}}{2}\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

$$\left(-\frac{5y^{2}}{2}\right)x^{-2}$$