Solve frac{z^frac{5{4}}z^-frac{1{4}}}{z^frac{1{3}}}

$$\frac { z ^ { \frac { 5 } { 4 } } z ^ { - \frac { 1 } { 4 } } } { z ^ { \frac { 1 } { 3 } } }$$

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{5}{4} and -\frac{1}{4} to get 1.

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

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

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

Differentiate w.r.t. z

$$\frac{2}{3\sqrt[3]{z}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{5}{4} and -\frac{1}{4} to get 1.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{\frac{2}{3}})$$

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

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

Subtract 1 from \frac{2}{3}.

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