Solve (w^frac{1{3}})^-frac{1{6}}

$$( w ^ { \frac { 1 } { 3 } } ) ^ { - \frac { 1 } { 6 } }$$

Differentiate w.r.t. w

$$-\frac{1}{18w^{\frac{19}{18}}}$$

Steps Using Chain Rule

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).

$$-\frac{1}{6}\left(\sqrt[3]{w}\right)^{-\frac{1}{6}-1}\frac{\mathrm{d}}{\mathrm{d}w}(\sqrt[3]{w})$$

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}.

$$-\frac{1}{6}\left(\sqrt[3]{w}\right)^{-\frac{7}{6}}\times \frac{1}{3}w^{\frac{1}{3}-1}$$

Simplify.

$$-\frac{1}{18}w^{-\frac{2}{3}}\left(\sqrt[3]{w}\right)^{-\frac{7}{6}}$$

Evaluate

$$\frac{1}{\sqrt[18]{w}}$$