Solve (mn^2/2nm^3*2m^0n^3)=

$$( \frac { m n ^ { 2 } } { 2 n m ^ { 3 } \cdot 2 m ^ { 0 } n ^ { 3 } } ) =$$

Evaluate

$$\frac{1}{4\left(mn\right)^{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{mn^{2}}{2n^{4}m^{3}\times 2m^{0}}$$

To multiply powers of the same base, add their exponents. Add 3 and 0 to get 3.

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

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

$$\frac{1}{2\times 2m^{2}n^{2}}$$

Multiply 2 and 2 to get 4.

$$\frac{1}{4m^{2}n^{2}}$$

Differentiate w.r.t. n

$$-\frac{1}{2m^{2}n^{3}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{4m^{2}n^{3}}n^{1})$$

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}.

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

Do the arithmetic.

$$\frac{1}{4m^{2}n^{3}}n^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{4m^{2}n^{3}}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{4m^{2}n^{3}}$$