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

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

Differentiate w.r.t. a

$$-\frac{1}{6a^{\frac{7}{6}}}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(a^{-\frac{1}{4}}\right)^{\frac{2}{3}})$$

To raise a power to another power, multiply the exponents. Multiply -\frac{1}{4} and \frac{2}{3} to get -\frac{1}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{-\frac{1}{6}})$$

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

$$-\frac{1}{6}a^{-\frac{1}{6}-1}$$

Subtract 1 from -\frac{1}{6}.

$$-\frac{1}{6}a^{-\frac{7}{6}}$$

Evaluate

$$\frac{1}{\sqrt[6]{a}}$$

Solution Steps

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

$$\left(a^{-\frac{1}{4}}\right)^{\frac{2}{3}}$$