Solve 4x*2/3x^-frac{1{3}}=

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

Evaluate

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

Solution Steps

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

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

Multiply 4 and \frac{2}{3} to get \frac{8}{3}.

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

Differentiate w.r.t. x

$$\frac{16}{9\sqrt[3]{x}}$$

Steps Using Definition of a Derivative

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

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

Multiply 4 and \frac{2}{3} to get \frac{8}{3}.

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

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

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

Multiply \frac{2}{3} times \frac{8}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\frac{16}{9}x^{\frac{2}{3}-1}$$

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

$$\frac{16}{9}x^{-\frac{1}{3}}$$