Solve -2/9x^3div2/3x

$$- \frac{ 2 }{ 9 } { x }^{ 3 } \div \frac{ 2 }{ 3 } x$$

Evaluate

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

Solution Steps

Divide -\frac{2}{9}x^{3} by \frac{2}{3} by multiplying -\frac{2}{9}x^{3} by the reciprocal of \frac{2}{3}.

$$\frac{-\frac{2}{9}x^{3}\times 3}{2}x$$

Multiply -\frac{2}{9} and 3 to get -\frac{2}{3}.

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

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

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

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

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

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

Divide -\frac{2}{9}x^{3} by \frac{2}{3} by multiplying -\frac{2}{9}x^{3} by the reciprocal of \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-\frac{2}{9}x^{3}\times 3}{2}x)$$

Multiply -\frac{2}{9} and 3 to get -\frac{2}{3}.

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

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

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

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

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

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

$$4\left(-\frac{1}{3}\right)x^{4-1}$$

Multiply 4 times -\frac{1}{3}.

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

Subtract 1 from 4.

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