Solve m^4n^-3*mn/(n^2)^-1

$$\frac { m ^ { 4 } n ^ { - 3 } \cdot m n } { ( n ^ { 2 } ) ^ { - 1 } }$$

Evaluate

$$m^{5}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

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

To multiply powers of the same base, add their exponents. Add -3 and 1 to get -2.

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

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

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

Cancel out n^{-2} in both numerator and denominator.

$$m^{5}$$

Differentiate w.r.t. m

$$5m^{4}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

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

To multiply powers of the same base, add their exponents. Add -3 and 1 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{m^{5}n^{-2}}{\left(n^{2}\right)^{-1}})$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{m^{5}n^{-2}}{n^{-2}})$$

Cancel out n^{-2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}m}(m^{5})$$

The derivative of ax^{n} is nax^{n-1}.

$$5m^{5-1}$$

Subtract 1 from 5.

$$5m^{4}$$