Solve 3x^frac{1{3}}div6x^frac{2{3}}

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

Evaluate

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

Solution Steps

Divide 3x^{\frac{1}{3}} by 6 to get \frac{1}{2}x^{\frac{1}{3}}.

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

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

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

Calculate x to the power of 1 and get x.

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

Differentiate w.r.t. x

$$\frac{1}{2} = 0.5$$

Steps Using Definition of a Derivative

Divide 3x^{\frac{1}{3}} by 6 to get \frac{1}{2}x^{\frac{1}{3}}.

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

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

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

Calculate x to the power of 1 and get x.

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

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

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

Subtract 1 from 1.

$$\frac{1}{2}x^{0}$$

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

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

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

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