Solve frac{3^{2}x^{4}y^2}{6^3x^2y^3}

$$\frac { 3 ^ { 2 } x ^ { 4 } y ^ { 2 } } { 6 ^ { 3 } x ^ { 2 } y ^ { 3 } }$$

Evaluate

$$\frac{x^{2}}{24y}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{3^{2}x^{4}y^{2}}{6^{3}x^{2}y^{3}}$$

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

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

Subtract 2 from 4.

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

Subtract 3 from 2.

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

Reduce the fraction \frac{9}{216} to lowest terms by extracting and canceling out 9.

$$\frac{1}{24}x^{2}\times \frac{1}{y}$$

Differentiate w.r.t. x

$$\frac{x}{12y}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9y^{2}}{216y^{3}}x^{4-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{24y}x^{2})$$

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}.

$$2\times \frac{1}{24y}x^{2-1}$$

Do the arithmetic.

$$\frac{1}{12y}x^{1}$$

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

$$\frac{1}{12y}x$$