Solve 2mn^2/3m^2n^4

$$\frac { 2 m n ^ { 2 } } { 3 m ^ { 2 } n ^ { 4 } }$$

Evaluate

$$\frac{2}{3mn^{2}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{2^{1}m^{1}n^{2}}{3^{1}m^{2}n^{4}}$$

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

$$\frac{2^{1}}{3^{1}}m^{1-2}n^{2-4}$$

Subtract 2 from 1.

$$\frac{2^{1}}{3^{1}}\times \frac{1}{m}n^{2-4}$$

Subtract 4 from 2.

$$\frac{2^{1}}{3^{1}}\times \frac{1}{m}n^{-2}$$

Divide 2 by 3.

$$\frac{2}{3}\times \frac{1}{m}\times \frac{1}{n^{2}}$$

Differentiate w.r.t. m

$$-\frac{2}{3\left(mn\right)^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{2n^{2}}{3n^{4}}m^{1-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{2}{3n^{2}}\times \frac{1}{m})$$

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}.

$$-\frac{2}{3n^{2}}m^{-1-1}$$

Do the arithmetic.

$$\left(-\frac{2}{3n^{2}}\right)m^{-2}$$