Solve (5m^-2n^3/10n^-1m^2)

$$( \frac { 5 m ^ { - 2 } n ^ { 3 } } { 10 n ^ { - 1 } m ^ { 2 } } )$$

Evaluate

$$\frac{\left(\frac{n}{m}\right)^{4}}{2}$$

Solution Steps

Cancel out 5 in both numerator and denominator.

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

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

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

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

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

Differentiate w.r.t. m

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{5n^{3}n}{10}m^{-2-2})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-2n^{4}\right)m^{-5}$$