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

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

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 3 from 5.

$$\frac{8^{1}}{12^{1}}x^{2}y^{1-4}$$

Subtract 4 from 1.

$$\frac{8^{1}}{12^{1}}x^{2}y^{-3}$$

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

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

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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

For any term t, t^{1}=t.

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