Solve (4x^2y^7/3x^4y^2)

$$( \frac { 4 x ^ { 2 } y ^ { 7 } } { 3 x ^ { 4 } y ^ { 2 } } )$$

Evaluate

$$\frac{4y^{5}}{3x^{2}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{2}y^{7}}{3^{1}x^{4}y^{2}}$$

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

$$\frac{4^{1}}{3^{1}}x^{2-4}y^{7-2}$$

Subtract 4 from 2.

$$\frac{4^{1}}{3^{1}}x^{-2}y^{7-2}$$

Subtract 2 from 7.

$$\frac{4^{1}}{3^{1}}\times \frac{1}{x^{2}}y^{5}$$

Divide 4 by 3.

$$\frac{4}{3}\times \frac{1}{x^{2}}y^{5}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y^{7}}{3y^{2}}x^{2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y^{5}}{3}x^{-2})$$

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}.

$$-2\times \frac{4y^{5}}{3}x^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{8y^{5}}{3}\right)x^{-3}$$