Solve frac{a^frac{3{2}}a^-frac{1{4}}}{a^frac{1{3}}}

$$\frac { a ^ { \frac { 3 } { 2 } } a ^ { - \frac { 1 } { 4 } } } { a ^ { \frac { 1 } { 3 } } }$$

Evaluate

$$a^{\frac{11}{12}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{3}{2} and -\frac{1}{4} to get \frac{5}{4}.

$$\frac{a^{\frac{5}{4}}}{a^{\frac{1}{3}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{1}{3} from \frac{5}{4} to get \frac{11}{12}.

$$a^{\frac{11}{12}}$$

Differentiate w.r.t. a

$$\frac{11}{12\sqrt[12]{a}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{3}{2} and -\frac{1}{4} to get \frac{5}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{\frac{5}{4}}}{a^{\frac{1}{3}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{1}{3} from \frac{5}{4} to get \frac{11}{12}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{11}{12}})$$

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

$$\frac{11}{12}a^{\frac{11}{12}-1}$$

Subtract 1 from \frac{11}{12}.

$$\frac{11}{12}a^{-\frac{1}{12}}$$