Solve (x^frac{7{5}})^-frac{5{3}}

$$( x ^ { \frac { 7 } { 5 } } ) ^ { - \frac { 5 } { 3 } }$$

Evaluate

$$\frac{1}{x^{\frac{7}{3}}}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\left(x^{\frac{7}{5}}\right)^{-\frac{5}{3}}$$

To raise a power to another power, multiply the exponents.

$$x^{\frac{7}{5}\left(-\frac{5}{3}\right)}$$

Multiply \frac{7}{5} times -\frac{5}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\frac{1}{x^{\frac{7}{3}}}$$

Differentiate w.r.t. x

$$-\frac{7}{3x^{\frac{10}{3}}}$$

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).

$$-\frac{5}{3}\left(x^{\frac{7}{5}}\right)^{-\frac{5}{3}-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{7}{5}})$$

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}.

$$-\frac{5}{3}\left(x^{\frac{7}{5}}\right)^{-\frac{8}{3}}\times \frac{7}{5}x^{\frac{7}{5}-1}$$

Simplify.

$$-\frac{7}{3}x^{\frac{2}{5}}\left(x^{\frac{7}{5}}\right)^{-\frac{8}{3}}$$