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

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

Differentiate w.r.t. a

$$\frac{1}{2\sqrt{a}}$$

Steps Using Chain Rule

Multiply a^{-\frac{1}{3}} and a^{\frac{1}{3}} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{-\frac{1}{2}}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

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

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

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

Simplify.

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

Evaluate

$$\sqrt{a}$$