Solve 6m^2n^4/12m^4n^5

$$\frac { 6 m ^ { 2 } n ^ { 4 } } { 12 m ^ { 4 } n ^ { 5 } }$$

Evaluate

$$\frac{1}{2nm^{2}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{6^{1}m^{2}n^{4}}{12^{1}m^{4}n^{5}}$$

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

$$\frac{6^{1}}{12^{1}}m^{2-4}n^{4-5}$$

Subtract 4 from 2.

$$\frac{6^{1}}{12^{1}}m^{-2}n^{4-5}$$

Subtract 5 from 4.

$$\frac{6^{1}}{12^{1}}\times \frac{1}{m^{2}}\times \frac{1}{n}$$

Reduce the fraction \frac{6}{12} to lowest terms by extracting and canceling out 6.

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

Differentiate w.r.t. m

$$-\frac{1}{nm^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6n^{4}}{12n^{5}}m^{2-4})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{1}{n}\right)m^{-3}$$