Solve 14x^2y^7/6x^5y^4

$$\frac { 14 x ^ { 2 } y ^ { 7 } } { 6 x ^ { 5 } y ^ { 4 } }$$

Evaluate

$$\frac{7\times \left(\frac{y}{x}\right)^{3}}{3}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{14^{1}x^{2}y^{7}}{6^{1}x^{5}y^{4}}$$

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

$$\frac{14^{1}}{6^{1}}x^{2-5}y^{7-4}$$

Subtract 5 from 2.

$$\frac{14^{1}}{6^{1}}x^{-3}y^{7-4}$$

Subtract 4 from 7.

$$\frac{14^{1}}{6^{1}}\times \frac{1}{x^{3}}y^{3}$$

Reduce the fraction \frac{14}{6} to lowest terms by extracting and canceling out 2.

$$\frac{7}{3}\times \frac{1}{x^{3}}y^{3}$$

Differentiate w.r.t. x

$$-\frac{7y^{3}}{x^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{14y^{7}}{6y^{4}}x^{2-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7y^{3}}{3}x^{-3})$$

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}.

$$-3\times \frac{7y^{3}}{3}x^{-3-1}$$

Do the arithmetic.

$$\left(-7y^{3}\right)x^{-4}$$