Solve (frac{x^{-3/5}x^{2/3}x^-3}{x^frac{2{7}}x^5/2})^-frac{2{5}}

$$( \frac { x ^ { - 3 / 5 } x ^ { 2 / 3 } x ^ { - 3 } } { x ^ { \frac { 2 } { 7 } } x ^ { 5 / 2 } } ) ^ { - \frac { 2 } { 5 } }$$

Evaluate

$$x^{\frac{1201}{525}}$$

Steps for Raising Power to Another Power

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

$$\left(\frac{x^{\frac{1}{15}}x^{-3}}{x^{\frac{2}{7}}x^{\frac{5}{2}}}\right)^{-\frac{2}{5}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{15} and -3 to get -\frac{44}{15}.

$$\left(\frac{x^{-\frac{44}{15}}}{x^{\frac{2}{7}}x^{\frac{5}{2}}}\right)^{-\frac{2}{5}}$$

To multiply powers of the same base, add their exponents. Add \frac{2}{7} and \frac{5}{2} to get \frac{39}{14}.

$$\left(\frac{x^{-\frac{44}{15}}}{x^{\frac{39}{14}}}\right)^{-\frac{2}{5}}$$

Rewrite x^{\frac{39}{14}} as x^{-\frac{44}{15}}x^{\frac{1201}{210}}. Cancel out x^{-\frac{44}{15}} in both numerator and denominator.

$$\left(\frac{1}{x^{\frac{1201}{210}}}\right)^{-\frac{2}{5}}$$

To raise \frac{1}{x^{\frac{1201}{210}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{-\frac{2}{5}}}{\left(x^{\frac{1201}{210}}\right)^{-\frac{2}{5}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1201}{210} and -\frac{2}{5} to get -\frac{1201}{525}.

$$\frac{1^{-\frac{2}{5}}}{x^{-\frac{1201}{525}}}$$

Calculate 1 to the power of -\frac{2}{5} and get 1.

$$\frac{1}{x^{-\frac{1201}{525}}}$$

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

$$x^{-\frac{1201}{525}\left(-1\right)}$$

Multiply -\frac{1201}{525} times -1.

$$x^{\frac{1201}{525}}$$

Differentiate w.r.t. x

$$\frac{1201x^{\frac{676}{525}}}{525}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{\frac{1}{15}}x^{-3}}{x^{\frac{2}{7}}x^{\frac{5}{2}}}\right)^{-\frac{2}{5}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{15} and -3 to get -\frac{44}{15}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{-\frac{44}{15}}}{x^{\frac{2}{7}}x^{\frac{5}{2}}}\right)^{-\frac{2}{5}})$$

To multiply powers of the same base, add their exponents. Add \frac{2}{7} and \frac{5}{2} to get \frac{39}{14}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{-\frac{44}{15}}}{x^{\frac{39}{14}}}\right)^{-\frac{2}{5}})$$

Rewrite x^{\frac{39}{14}} as x^{-\frac{44}{15}}x^{\frac{1201}{210}}. Cancel out x^{-\frac{44}{15}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{1}{x^{\frac{1201}{210}}}\right)^{-\frac{2}{5}})$$

To raise \frac{1}{x^{\frac{1201}{210}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{-\frac{2}{5}}}{\left(x^{\frac{1201}{210}}\right)^{-\frac{2}{5}}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{1201}{210} and -\frac{2}{5} to get -\frac{1201}{525}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{-\frac{2}{5}}}{x^{-\frac{1201}{525}}})$$

Calculate 1 to the power of -\frac{2}{5} and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{-\frac{1201}{525}}})$$

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).

$$-\left(x^{-\frac{1201}{525}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{-\frac{1201}{525}})$$

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}.

$$-\left(x^{-\frac{1201}{525}}\right)^{-2}\left(-\frac{1201}{525}\right)x^{-\frac{1201}{525}-1}$$

Simplify.

$$\frac{1201}{525}x^{-\frac{1726}{525}}\left(x^{-\frac{1201}{525}}\right)^{-2}$$