Solve (X^(4/5))^(1/3)

$$( X ^ { ( 4 / 5 ) } ) ^ { ( 1 / 3 ) }$$

Differentiate w.r.t. X

$$\frac{4}{15X^{\frac{11}{15}}}$$

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).

$$\frac{1}{3}\left(X^{\frac{4}{5}}\right)^{\frac{1}{3}-1}\frac{\mathrm{d}}{\mathrm{d}X}(X^{\frac{4}{5}})$$

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}.

$$\frac{1}{3}\left(X^{\frac{4}{5}}\right)^{-\frac{2}{3}}\times \frac{4}{5}X^{\frac{4}{5}-1}$$

Simplify.

$$\frac{4}{15}X^{-\frac{1}{5}}\left(X^{\frac{4}{5}}\right)^{-\frac{2}{3}}$$

Evaluate

$$X^{\frac{4}{15}}$$