Solve {left(8xright)}^-frac{2{3}}

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

Evaluate

$$\frac{1}{4x^{\frac{2}{3}}}$$

Solution Steps

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

$$8^{-\frac{2}{3}}x^{-\frac{2}{3}}$$

Calculate 8 to the power of -\frac{2}{3} and get \frac{1}{4}.

$$\frac{1}{4}x^{-\frac{2}{3}}$$

Differentiate w.r.t. x

$$-\frac{1}{6x^{\frac{5}{3}}}$$

Steps Using Definition of a Derivative

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

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

Calculate 8 to the power of -\frac{2}{3} and get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4}x^{-\frac{2}{3}})$$

The derivative of ax^{n} is nax^{n-1}.

$$-\frac{2}{3}\times \frac{1}{4}x^{-\frac{2}{3}-1}$$

Multiply -\frac{2}{3} times \frac{1}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-\frac{1}{6}x^{-\frac{2}{3}-1}$$

Subtract 1 from -\frac{2}{3}.

$$-\frac{1}{6}x^{-\frac{5}{3}}$$