Solve x/3{frac{x^2/3}}

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

Evaluate

$$\frac{x^{3}}{9}$$

Solution Steps

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

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

Express x\times \frac{x^{2}}{3} as a single fraction.

$$\frac{\frac{xx^{2}}{3}}{3}$$

Express \frac{\frac{xx^{2}}{3}}{3} as a single fraction.

$$\frac{xx^{2}}{3\times 3}$$

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

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

Multiply 3 and 3 to get 9.

$$\frac{x^{3}}{9}$$

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

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

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

Express x\times \frac{x^{2}}{3} as a single fraction.

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

Express \frac{\frac{xx^{2}}{3}}{3} as a single fraction.

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

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

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

Multiply 3 and 3 to get 9.

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

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

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

Multiply 3 times \frac{1}{9}.

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

Subtract 1 from 3.

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