Solve (-8x^-frac{1{2}})^-frac{1{3}}

$$( - 8 x ^ { - \frac { 1 } { 2 } } ) ^ { - \frac { 1 } { 3 } }$$

Evaluate

$$-\frac{\sqrt[6]{x}}{2}$$

Solution Steps

Expand \left(-8x^{-\frac{1}{2}}\right)^{-\frac{1}{3}}.

$$\left(-8\right)^{-\frac{1}{3}}\left(x^{-\frac{1}{2}}\right)^{-\frac{1}{3}}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{1}{2} and -\frac{1}{3} to get \frac{1}{6}.

$$\left(-8\right)^{-\frac{1}{3}}x^{\frac{1}{6}}$$

Calculate -8 to the power of -\frac{1}{3} and get -\frac{1}{2}.

$$-\frac{1}{2}x^{\frac{1}{6}}$$

Differentiate w.r.t. x

$$-\frac{1}{12x^{\frac{5}{6}}}$$

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).

$$-\frac{1}{3}\left(-8x^{-\frac{1}{2}}\right)^{-\frac{1}{3}-1}\frac{\mathrm{d}}{\mathrm{d}x}(-8x^{-\frac{1}{2}})$$

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}.

$$-\frac{1}{3}\left(-8x^{-\frac{1}{2}}\right)^{-\frac{4}{3}}\left(-\frac{1}{2}\right)\left(-8\right)x^{-\frac{1}{2}-1}$$

Simplify.

$$-\frac{4}{3}x^{-\frac{3}{2}}\left(-8x^{-\frac{1}{2}}\right)^{-\frac{4}{3}}$$