Solve frac{m^{-1}n^{-2}p^-8}{m^4n^5p^3}

$$\frac { m ^ { - 1 } n ^ { - 2 } p ^ { - 8 } } { m ^ { 4 } n ^ { 5 } p ^ { 3 } }$$

Differentiate w.r.t. m

$$-\frac{5}{m^{6}n^{7}p^{11}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{1}{n^{2}p^{8}p^{3}n^{5}}m^{-1-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{1}{n^{7}p^{11}}m^{-5})$$

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}.

$$-5\times \frac{1}{n^{7}p^{11}}m^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{5}{n^{7}p^{11}}\right)m^{-6}$$

Evaluate

$$\frac{1}{m^{5}n^{7}p^{11}}$$

Solution Steps

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

$$\frac{1}{m^{5}n^{7}p^{11}}$$