Solve x^2/3x^1/3quad2/3

$$x ^ { 2 / 3 } x ^ { 1 / 3 } \quad \frac { 2 } { 3 }$$

Evaluate

$$\frac{2x}{3}$$

Solution Steps

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

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

Calculate x to the power of 1 and get x.

$$x\times \frac{2}{3}$$

Differentiate w.r.t. x

$$\frac{2}{3} = 0.6666666666666666$$

Steps Using Definition of a Derivative

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

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

Calculate x to the power of 1 and get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x\times \frac{2}{3})$$

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

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

Subtract 1 from 1.

$$\frac{2}{3}x^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{2}{3}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{2}{3}$$