Solve frac{-6m^{5}n^{2}q^-1}{36m^-2n^4q^-1}

$$\frac { - 6 m ^ { 5 } n ^ { 2 } q ^ { - 1 } } { 36 m ^ { - 2 } n ^ { 4 } q ^ { - 1 } }$$

Evaluate

$$-\frac{m^{7}}{6n^{2}}$$

Solution Steps

Cancel out 6\times \frac{1}{q}n^{2} in both numerator and denominator.

$$\frac{-m^{5}}{6m^{-2}n^{2}}$$

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

$$\frac{-m^{7}}{6n^{2}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{m^{7}}{-6n^{2}}$$

Differentiate w.r.t. m

$$-\frac{7m^{6}}{6n^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{\left(-\frac{6n^{2}}{q}\right)q}{36n^{4}}m^{5-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\left(-\frac{1}{6n^{2}}\right)m^{7})$$

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}.

$$7\left(-\frac{1}{6n^{2}}\right)m^{7-1}$$

Do the arithmetic.

$$\left(-\frac{7}{6n^{2}}\right)m^{6}$$