Solve (z^frac{1{3}})^-2

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

Differentiate w.r.t. z

$$-\frac{2}{3z^{\frac{5}{3}}}$$

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).

$$-2\left(\sqrt[3]{z}\right)^{-2-1}\frac{\mathrm{d}}{\mathrm{d}z}(\sqrt[3]{z})$$

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}.

$$-2\left(\sqrt[3]{z}\right)^{-3}\times \frac{1}{3}z^{\frac{1}{3}-1}$$

Simplify.

$$-\frac{2}{3}z^{-\frac{2}{3}}\left(\sqrt[3]{z}\right)^{-3}$$

Evaluate

$$\frac{1}{z^{\frac{2}{3}}}$$