Solve 12xy^2/3x^4y^2

$$\frac { 12 x y ^ { 2 } } { 3 x ^ { 4 } y ^ { 2 } }$$

Evaluate

$$\frac{4}{x^{3}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 4 from 1.

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

Subtract 2 from 2.

$$\frac{12^{1}}{3^{1}}\times \frac{1}{x^{3}}y^{0}$$

For any number a except 0, a^{0}=1.

$$\frac{12^{1}}{3^{1}}\times \frac{1}{x^{3}}$$

Divide 12 by 3.

$$4\times \frac{1}{x^{3}}$$

Differentiate w.r.t. x

$$-\frac{12}{x^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12y^{2}}{3y^{2}}x^{1-4})$$

Do the arithmetic.

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

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}.

$$-3\times 4x^{-3-1}$$

Do the arithmetic.

$$-12x^{-4}$$