Solve (frac{x^{-3}x^{2/3}x^-3}{x^3/4x^5/2})^frac{-2{5}}

$$( \frac { x ^ { - 3 } x ^ { 2 / 3 } x ^ { - 3 } } { x ^ { 3 / 4 } x ^ { 5 / 2 } } ) ^ { \frac { - 2 } { 5 } }$$

Differentiate w.r.t. x

$$\frac{103x^{\frac{73}{30}}}{30}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add -3 and \frac{2}{3} to get -\frac{7}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{-\frac{7}{3}}x^{-3}}{x^{\frac{3}{4}}x^{\frac{5}{2}}}\right)^{\frac{-2}{5}})$$

To multiply powers of the same base, add their exponents. Add -\frac{7}{3} and -3 to get -\frac{16}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{-\frac{16}{3}}}{x^{\frac{3}{4}}x^{\frac{5}{2}}}\right)^{\frac{-2}{5}})$$

To multiply powers of the same base, add their exponents. Add \frac{3}{4} and \frac{5}{2} to get \frac{13}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{-\frac{16}{3}}}{x^{\frac{13}{4}}}\right)^{\frac{-2}{5}})$$

Rewrite x^{\frac{13}{4}} as x^{-\frac{16}{3}}x^{\frac{103}{12}}. Cancel out x^{-\frac{16}{3}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{1}{x^{\frac{103}{12}}}\right)^{\frac{-2}{5}})$$

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{1}{x^{\frac{103}{12}}}\right)^{-\frac{2}{5}})$$

To raise \frac{1}{x^{\frac{103}{12}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{-\frac{2}{5}}}{\left(x^{\frac{103}{12}}\right)^{-\frac{2}{5}}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{103}{12} and -\frac{2}{5} to get -\frac{103}{30}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{-\frac{2}{5}}}{x^{-\frac{103}{30}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{-\frac{103}{30}}})$$

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).

$$-\left(x^{-\frac{103}{30}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{-\frac{103}{30}})$$

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}.

$$-\left(x^{-\frac{103}{30}}\right)^{-2}\left(-\frac{103}{30}\right)x^{-\frac{103}{30}-1}$$

Simplify.

$$\frac{103}{30}x^{-\frac{133}{30}}\left(x^{-\frac{103}{30}}\right)^{-2}$$

Evaluate

$$x^{\frac{103}{30}}$$