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

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

Evaluate

$$-\frac{1}{2yx^{2}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 3 from 1.

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

Subtract 2 from 1.

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

Reduce the fraction \frac{2}{-4} to lowest terms by extracting and canceling out 2.

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

Differentiate w.r.t. x

$$\frac{1}{yx^{3}}$$

Solution Steps

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

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

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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