Solve 4x^4y/8x^5y^3

$$\frac { 4 x ^ { 4 } y } { 8 x ^ { 5 } y ^ { 3 } }$$

Evaluate

$$\frac{1}{2xy^{2}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{4}y^{1}}{8^{1}x^{5}y^{3}}$$

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

$$\frac{4^{1}}{8^{1}}x^{4-5}y^{1-3}$$

Subtract 5 from 4.

$$\frac{4^{1}}{8^{1}}\times \frac{1}{x}y^{1-3}$$

Subtract 3 from 1.

$$\frac{4^{1}}{8^{1}}\times \frac{1}{x}y^{-2}$$

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

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

Differentiate w.r.t. x

$$-\frac{1}{2\left(xy\right)^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y}{8y^{3}}x^{4-5})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{1}{2y^{2}}\right)x^{-2}$$