Solve frac{3x^-frac{2{3}}}{6x^frac{1{3}}}

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

Evaluate

$$\frac{1}{2x}$$

Solution Steps

Cancel out 3 in both numerator and denominator.

$$\frac{x^{-\frac{2}{3}}}{2\sqrt[3]{x}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

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

Calculate x to the power of 1 and get x.

$$\frac{1}{2x}$$

Differentiate w.r.t. x

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

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{6}x^{-\frac{2}{3}-\frac{1}{3}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

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