Solve 5/12x^4y^2/-frac{1{6}x^3y}

$$\frac{ \frac{ 5 }{ 12 } { x }^{ 4 } { y }^{ 2 } }{ - \frac{ 1 }{ 6 } { x }^{ 3 } y }$$

Evaluate

$$-\frac{5xy}{2}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(\frac{5}{12}\right)^{1}x^{4}y^{2}}{\left(-\frac{1}{6}\right)^{1}x^{3}y^{1}}$$

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

$$\frac{\left(\frac{5}{12}\right)^{1}}{\left(-\frac{1}{6}\right)^{1}}x^{4-3}y^{2-1}$$

Subtract 3 from 4.

$$\frac{\left(\frac{5}{12}\right)^{1}}{\left(-\frac{1}{6}\right)^{1}}x^{1}y^{2-1}$$

Subtract 1 from 2.

$$\frac{\left(\frac{5}{12}\right)^{1}}{\left(-\frac{1}{6}\right)^{1}}xy^{1}$$

Divide \frac{5}{12} by -\frac{1}{6} by multiplying \frac{5}{12} by the reciprocal of -\frac{1}{6}.

$$-\frac{5}{2}xy$$

Differentiate w.r.t. x

$$-\frac{5y}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5y^{2}}{12\left(-\frac{y}{6}\right)}x^{4-3})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{5y}{2}\right)x^{0}$$

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

$$\left(-\frac{5y}{2}\right)\times 1$$

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

$$-\frac{5y}{2}$$