Solve (frac{{left({n}^frac{1{2}}right)}^frac{1{3}}}{{n}^-frac{4{3}}n^2})

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

Differentiate w.r.t. n

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

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{n^{\frac{1}{6}}}{n^{-\frac{4}{3}}n^{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{n^{\frac{1}{6}}}{n^{\frac{2}{3}}})$$

Rewrite n^{\frac{2}{3}} as n^{\frac{1}{6}}n^{\frac{1}{2}}. Cancel out n^{\frac{1}{6}} in both numerator and denominator.

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

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).

$$-\left(\sqrt{n}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}n}(\sqrt{n})$$

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}.

$$-\left(\sqrt{n}\right)^{-2}\times \frac{1}{2}n^{\frac{1}{2}-1}$$

Simplify.

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

Evaluate

$$\frac{1}{\sqrt{n}}$$