Solve -frac{n^{4}}{m}divn^2/m^2*m^2/n^2

$$- \frac { n ^ { 4 } } { m } \div \frac { n ^ { 2 } } { m ^ { 2 } } \cdot \frac { m ^ { 2 } } { n ^ { 2 } }$$

Evaluate

$$-m^{3}$$

Solution Steps

Divide -\frac{n^{4}}{m} by \frac{n^{2}}{m^{2}} by multiplying -\frac{n^{4}}{m} by the reciprocal of \frac{n^{2}}{m^{2}}.

$$\frac{\left(-\frac{n^{4}}{m}\right)m^{2}}{n^{2}}\times \frac{m^{2}}{n^{2}}$$

Express \left(-\frac{n^{4}}{m}\right)m^{2} as a single fraction.

$$\frac{\frac{-n^{4}m^{2}}{m}}{n^{2}}\times \frac{m^{2}}{n^{2}}$$

Cancel out m in both numerator and denominator.

$$\frac{-mn^{4}}{n^{2}}\times \frac{m^{2}}{n^{2}}$$

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

$$-mn^{2}\times \frac{m^{2}}{n^{2}}$$

Express m\times \frac{m^{2}}{n^{2}} as a single fraction.

$$-\frac{mm^{2}}{n^{2}}n^{2}$$

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

$$-\frac{m^{3}}{n^{2}}n^{2}$$

Cancel out n^{2} and n^{2}.

$$-m^{3}$$

Differentiate w.r.t. m

$$-3m^{2}$$

Steps Using Definition of a Derivative

Divide -\frac{n^{4}}{m} by \frac{n^{2}}{m^{2}} by multiplying -\frac{n^{4}}{m} by the reciprocal of \frac{n^{2}}{m^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{\left(-\frac{n^{4}}{m}\right)m^{2}}{n^{2}}\times \frac{m^{2}}{n^{2}})$$

Express \left(-\frac{n^{4}}{m}\right)m^{2} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{\frac{-n^{4}m^{2}}{m}}{n^{2}}\times \frac{m^{2}}{n^{2}})$$

Cancel out m in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{-mn^{4}}{n^{2}}\times \frac{m^{2}}{n^{2}})$$

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

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

Express m\times \frac{m^{2}}{n^{2}} as a single fraction.

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

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

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

Cancel out n^{2} and n^{2}.

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

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

$$3\left(-1\right)m^{3-1}$$

Multiply 3 times -1.

$$-3m^{3-1}$$

Subtract 1 from 3.

$$-3m^{2}$$