Solve (-30m^{12}m^7z^4)div(6m^10n^7z)

$$( - 30 m ^ { 12 } m ^ { 7 } z ^ { 4 } ) \div ( 6 m ^ { 10 } n ^ { 7 } z )$$

Evaluate

$$-\frac{5z^{3}m^{9}}{n^{7}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 12 and 7 to get 19.

$$\frac{-30m^{19}z^{4}}{6m^{10}n^{7}z}$$

Cancel out 6zm^{10} in both numerator and denominator.

$$\frac{-5z^{3}m^{9}}{n^{7}}$$

Differentiate w.r.t. m

$$-\frac{45z^{3}m^{8}}{n^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\left(-\frac{30z^{4}m^{7}}{6zn^{7}}\right)m^{12-10})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\left(-\frac{5z^{3}m^{7}}{n^{7}}\right)m^{2})$$

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}.

$$2\left(-\frac{5z^{3}m^{7}}{n^{7}}\right)m^{2-1}$$

Do the arithmetic.

$$\left(-\frac{10z^{3}m^{7}}{n^{7}}\right)m^{1}$$

For any term t, t^{1}=t.

$$\left(-\frac{10z^{3}m^{7}}{n^{7}}\right)m$$