Solve (x^1/3)^2/(x^2)^5/3

$$\frac { ( x ^ { 1 / 3 } ) ^ { 2 } } { ( x ^ { 2 } ) ^ { 5 / 3 } }$$

Differentiate w.r.t. x

$$-\frac{8}{3x^{\frac{11}{3}}}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{2}{3}}}{\left(x^{2}\right)^{\frac{5}{3}}})$$

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{5}{3} to get \frac{10}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{2}{3}}}{x^{\frac{10}{3}}})$$

Rewrite x^{\frac{10}{3}} as x^{\frac{2}{3}}x^{\frac{8}{3}}. Cancel out x^{\frac{2}{3}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{\frac{8}{3}}})$$

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).

$$-\left(x^{\frac{8}{3}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{8}{3}})$$

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}.

$$-\left(x^{\frac{8}{3}}\right)^{-2}\times \frac{8}{3}x^{\frac{8}{3}-1}$$

Simplify.

$$-\frac{8}{3}x^{\frac{5}{3}}\left(x^{\frac{8}{3}}\right)^{-2}$$

Evaluate

$$\frac{1}{x^{\frac{8}{3}}}$$