Solve frac{45m^{-6}p^{2}v^12}{15m^-2p^8v^-4}

$$\frac { 45 m ^ { - 6 } p ^ { 2 } v ^ { 12 } } { 15 m ^ { - 2 } p ^ { 8 } v ^ { - 4 } }$$

Evaluate

$$\frac{3v^{16}}{m^{4}p^{6}}$$

Solution Steps

Cancel out 15p^{2} in both numerator and denominator.

$$\frac{3m^{-6}v^{12}}{v^{-4}m^{-2}p^{6}}$$

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

$$\frac{3m^{-6}v^{16}}{m^{-2}p^{6}}$$

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

$$\frac{3v^{16}}{m^{4}p^{6}}$$

Differentiate w.r.t. m

$$-\frac{12v^{16}}{m^{5}p^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{45p^{2}v^{12}v^{4}}{15p^{8}}m^{-6-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{3v^{16}}{p^{6}}m^{-4})$$

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}.

$$-4\times \frac{3v^{16}}{p^{6}}m^{-4-1}$$

Do the arithmetic.

$$\left(-\frac{12v^{16}}{p^{6}}\right)m^{-5}$$