Solve frac{z^frac{1{3}}}{z^-frac{3{2}}z^frac{1{2}}}

$$\frac { z ^ { \frac { 1 } { 3 } } } { z ^ { - \frac { 3 } { 2 } } z ^ { \frac { 1 } { 2 } } }$$

Differentiate w.r.t. z

$$\frac{4\sqrt[3]{z}}{3}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{\frac{1}{3}}}{z^{-1}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -1 from \frac{1}{3} to get \frac{4}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{\frac{4}{3}})$$

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

$$\frac{4}{3}z^{\frac{4}{3}-1}$$

Subtract 1 from \frac{4}{3}.

$$\frac{4}{3}\sqrt[3]{z}$$

Evaluate

$$z^{\frac{4}{3}}$$