Solve -30m^10n^-10/15m^-10n^10

$$\frac { - 30 m ^ { 10 } n ^ { - 10 } } { 15 m ^ { - 10 } n ^ { 10 } }$$

Evaluate

$$-2\times \left(\frac{m}{n}\right)^{20}$$

Solution Steps

Cancel out 15 in both numerator and denominator.

$$\frac{-2n^{-10}m^{10}}{m^{-10}n^{10}}$$

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

$$\frac{-2n^{-10}m^{20}}{n^{10}}$$

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

$$\frac{-2m^{20}}{n^{20}}$$

Differentiate w.r.t. m

$$-\frac{40m^{19}}{n^{20}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\left(-\frac{\frac{30}{n^{10}}}{15n^{10}}\right)m^{10-\left(-10\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\left(-\frac{2}{n^{20}}\right)m^{20})$$

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}.

$$20\left(-\frac{2}{n^{20}}\right)m^{20-1}$$

Do the arithmetic.

$$\left(-\frac{40}{n^{20}}\right)m^{19}$$