Solve 3xy/12x^2y^6

$$\frac { 3 x y } { 12 x ^ { 2 } y ^ { 6 } }$$

Evaluate

$$\frac{1}{4xy^{5}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 2 from 1.

$$\frac{3^{1}}{12^{1}}\times \frac{1}{x}y^{1-6}$$

Subtract 6 from 1.

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

Reduce the fraction \frac{3}{12} to lowest terms by extracting and canceling out 3.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y}{12y^{6}}x^{1-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4y^{5}}\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

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