Solve x/2*x/2*x/2

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

Evaluate

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

Solution Steps

Multiply \frac{x}{2} and \frac{x}{2} to get \left(\frac{x}{2}\right)^{2}.

$$\left(\frac{x}{2}\right)^{2}\times \frac{x}{2}$$

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

$$\left(\frac{x}{2}\right)^{3}$$

To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 2 to the power of 3 and get 8.

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

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

Multiply \frac{x}{2} and \frac{x}{2} to get \left(\frac{x}{2}\right)^{2}.

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

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

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

To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 2 to the power of 3 and get 8.

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

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

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

Multiply 3 times \frac{1}{8}.

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

Subtract 1 from 3.

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