Solve frac{{x}^frac{2{5}}{x}^-frac{1{3}}}{{x}^frac{2{3}}}

$$\frac{ { x }^{ \frac{ 2 }{ 5 } } { x }^{ - \frac{ 1 }{ 3 } } }{ { x }^{ \frac{ 2 }{ 3 } } }$$

Evaluate

$$\frac{1}{x^{\frac{3}{5}}}$$

Steps for Raising Power to Another Power

To multiply powers of the same base, add their exponents. Add \frac{2}{5} and -\frac{1}{3} to get \frac{1}{15}.

$$\frac{x^{\frac{1}{15}}}{x^{\frac{2}{3}}}$$

Rewrite x^{\frac{2}{3}} as x^{\frac{1}{15}}x^{\frac{3}{5}}. Cancel out x^{\frac{1}{15}} in both numerator and denominator.

$$\frac{1}{x^{\frac{3}{5}}}$$

Differentiate w.r.t. x

$$-\frac{3}{5x^{\frac{8}{5}}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add \frac{2}{5} and -\frac{1}{3} to get \frac{1}{15}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{1}{15}}}{x^{\frac{2}{3}}})$$

Rewrite x^{\frac{2}{3}} as x^{\frac{1}{15}}x^{\frac{3}{5}}. Cancel out x^{\frac{1}{15}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{\frac{3}{5}}})$$

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).

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

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}.

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

Simplify.

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