Solve frac{z^-frac{1{3}}}{z^-frac{5{6}}}

$$\frac { z ^ { - \frac { 1 } { 3 } } } { z ^ { - \frac { 5 } { 6 } } }$$

Differentiate w.r.t. z

$$\frac{1}{2\sqrt{z}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{1}z^{-\frac{1}{3}-\left(-\frac{5}{6}\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\sqrt{z})$$

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}.

$$\frac{1}{2}z^{\frac{1}{2}-1}$$

Do the arithmetic.

$$\frac{1}{2}z^{-\frac{1}{2}}$$

Evaluate

$$\sqrt{z}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{5}{6} from -\frac{1}{3} to get \frac{1}{2}.

$$z^{\frac{1}{2}}$$