Solve 1quad(x^frac{2{3}})^frac{1{3}}

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

Evaluate

$$x^{\frac{2}{9}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and \frac{1}{3} to get \frac{2}{9}.

$$1x^{\frac{2}{9}}$$

For any term t, t\times 1=t and 1t=t.

$$x^{\frac{2}{9}}$$

Differentiate w.r.t. x

$$\frac{2}{9x^{\frac{7}{9}}}$$

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).

$$\frac{1}{3}\left(x^{\frac{2}{3}}\right)^{\frac{1}{3}-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{2}{3}})$$

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}.

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

Simplify.

$$\frac{2}{9}x^{-\frac{1}{3}}\left(x^{\frac{2}{3}}\right)^{-\frac{2}{3}}$$