Solve frac{m^{2}n^{-1}p^-9}{m^-7n^8p^5}

$$\frac { m ^ { 2 } n ^ { - 1 } p ^ { - 9 } } { m ^ { - 7 } n ^ { 8 } p ^ { 5 } }$$

Evaluate

$$\frac{\left(\frac{m}{n}\right)^{9}}{p^{14}}$$

Solution Steps

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

$$\frac{p^{-9}\times \frac{1}{n}m^{9}}{p^{5}n^{8}}$$

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

$$\frac{m^{9}}{n^{9}p^{14}}$$

Differentiate w.r.t. m

$$\frac{9m^{8}}{n^{9}p^{14}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{1}{np^{9}p^{5}n^{8}}m^{2-\left(-7\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{1}{n^{9}p^{14}}m^{9})$$

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}.

$$9\times \frac{1}{n^{9}p^{14}}m^{9-1}$$

Do the arithmetic.

$$\frac{9}{n^{9}p^{14}}m^{8}$$