Solve frac{2^2m^3{n}^frac{1{2}}}{sqrt[3]{-64}m^2n}

$$\frac{ { 2 }^{ 2 } { m }^{ 3 } { n }^{ \frac{ 1 }{ 2 } } }{ \sqrt[ 3 ]{ -64 } { m }^{ 2 } n }$$

Differentiate w.r.t. m

$$-\frac{1}{\sqrt{n}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{4\sqrt{n}}{\sqrt[3]{-64}n}m^{3-2})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{1}{\sqrt{n}}\right)m^{0}$$

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

$$\left(-\frac{1}{\sqrt{n}}\right)\times 1$$

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

$$-\frac{1}{\sqrt{n}}$$

Evaluate

$$-\frac{m}{\sqrt{n}}$$