Solve frac{v^frac{1{3}}*v^frac{4{3}}}{v^frac{4{3}}}

$$\frac { v ^ { \frac { 1 } { 3 } } \cdot v ^ { \frac { 4 } { 3 } } } { v ^ { \frac { 4 } { 3 } } }$$

Evaluate

$$\sqrt[3]{v}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and \frac{4}{3} to get \frac{5}{3}.

$$\frac{v^{\frac{5}{3}}}{v^{\frac{4}{3}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{4}{3} from \frac{5}{3} to get \frac{1}{3}.

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

Differentiate w.r.t. v

$$\frac{1}{3v^{\frac{2}{3}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and \frac{4}{3} to get \frac{5}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v^{\frac{5}{3}}}{v^{\frac{4}{3}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{4}{3} from \frac{5}{3} to get \frac{1}{3}.

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

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

$$\frac{1}{3}v^{\frac{1}{3}-1}$$

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

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