Solve frac{m^{6}*n^{4}*m^4}{m^2*m^2*m^3*n}=

$$\frac { m ^ { 6 } \times n ^ { 4 } \times m ^ { 4 } } { m ^ { 2 } \times m ^ { 2 } \times m ^ { 3 } \times n } =$$

Evaluate

$$\left(mn\right)^{3}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 6 and 4 to get 10.

$$\frac{m^{10}n^{4}}{m^{2}m^{2}m^{3}n}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{m^{10}n^{4}}{m^{4}m^{3}n}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{m^{10}n^{4}}{m^{7}n}$$

Cancel out nm^{7} in both numerator and denominator.

$$m^{3}n^{3}$$

Differentiate w.r.t. m

$$3m^{2}n^{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{m^{4}n^{4}}{nm^{2}m^{3}}m^{6-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{n^{3}}{m}m^{4})$$

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}.

$$4\times \frac{n^{3}}{m}m^{4-1}$$

Do the arithmetic.

$$\frac{4n^{3}}{m}m^{3}$$