Solve a^1/5*a^2/3/a^3/5

$$\frac { a ^ { 1 / 5 } \times a ^ { 2 / 3 } } { a ^ { 3 / 5 } }$$

Evaluate

$$a^{\frac{4}{15}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{5} and \frac{2}{3} to get \frac{13}{15}.

$$\frac{a^{\frac{13}{15}}}{a^{\frac{3}{5}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{3}{5} from \frac{13}{15} to get \frac{4}{15}.

$$a^{\frac{4}{15}}$$

Differentiate w.r.t. a

$$\frac{4}{15a^{\frac{11}{15}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{5} and \frac{2}{3} to get \frac{13}{15}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{\frac{13}{15}}}{a^{\frac{3}{5}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{3}{5} from \frac{13}{15} to get \frac{4}{15}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{4}{15}})$$

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

$$\frac{4}{15}a^{\frac{4}{15}-1}$$

Subtract 1 from \frac{4}{15}.

$$\frac{4}{15}a^{-\frac{11}{15}}$$