Solve 8x^3y^5/12x^4y^3

$$\frac { 8 x ^ { 3 } y ^ { 5 } } { 12 x ^ { 4 } y ^ { 3 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{8^{1}x^{3}y^{5}}{12^{1}x^{4}y^{3}}$$

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

$$\frac{8^{1}}{12^{1}}x^{3-4}y^{5-3}$$

Subtract 4 from 3.

$$\frac{8^{1}}{12^{1}}\times \frac{1}{x}y^{5-3}$$

Subtract 3 from 5.

$$\frac{8^{1}}{12^{1}}\times \frac{1}{x}y^{2}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

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

Differentiate w.r.t. x

$$-\frac{2\times \left(\frac{y}{x}\right)^{2}}{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8y^{5}}{12y^{3}}x^{3-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2y^{2}}{3}\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

$$\left(-\frac{2y^{2}}{3}\right)x^{-2}$$