Solve 2/3x^2ydiv(-5/6xy^2)

$$\frac { 2 } { 3 } x ^ { 2 } y \div ( - \frac { 5 } { 6 } x y ^ { 2 } )$$

Evaluate

$$-\frac{4x}{5y}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(\frac{2}{3}\right)^{1}x^{2}y^{1}}{\left(-\frac{5}{6}\right)^{1}x^{1}y^{2}}$$

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

$$\frac{\left(\frac{2}{3}\right)^{1}}{\left(-\frac{5}{6}\right)^{1}}x^{2-1}y^{1-2}$$

Subtract 1 from 2.

$$\frac{\left(\frac{2}{3}\right)^{1}}{\left(-\frac{5}{6}\right)^{1}}x^{1}y^{1-2}$$

Subtract 2 from 1.

$$\frac{\left(\frac{2}{3}\right)^{1}}{\left(-\frac{5}{6}\right)^{1}}x\times \frac{1}{y}$$

Divide \frac{2}{3} by -\frac{5}{6} by multiplying \frac{2}{3} by the reciprocal of -\frac{5}{6}.

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

Differentiate w.r.t. x

$$-\frac{4}{5y}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2y}{3\left(-\frac{5y^{2}}{6}\right)}x^{2-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{4}{5y}\right)x^{1})$$

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}.

$$\left(-\frac{4}{5y}\right)x^{1-1}$$

Do the arithmetic.

$$\left(-\frac{4}{5y}\right)x^{0}$$

For any term t except 0, t^{0}=1.

$$\left(-\frac{4}{5y}\right)\times 1$$

For any term t, t\times 1=t and 1t=t.

$$-\frac{4}{5y}$$