Solve 1/2x^3-6x+7x-2/3x^3-x+1/4x^3

$$\frac { 1 } { 2 } x ^ { 3 } - 6 x + 7 x - \frac { 2 } { 3 } x ^ { 3 } - x + \frac { 1 } { 4 } x ^ { 3 }$$

Evaluate

$$\frac{x^{3}}{12}$$

Solution Steps

Combine -6x and 7x to get x.

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

Combine \frac{1}{2}x^{3} and -\frac{2}{3}x^{3} to get -\frac{1}{6}x^{3}.

$$-\frac{1}{6}x^{3}+x-x+\frac{1}{4}x^{3}$$

Combine x and -x to get 0.

$$-\frac{1}{6}x^{3}+\frac{1}{4}x^{3}$$

Combine -\frac{1}{6}x^{3} and \frac{1}{4}x^{3} to get \frac{1}{12}x^{3}.

$$\frac{1}{12}x^{3}$$

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

Combine -6x and 7x to get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}x^{3}+x-\frac{2}{3}x^{3}-x+\frac{1}{4}x^{3})$$

Combine \frac{1}{2}x^{3} and -\frac{2}{3}x^{3} to get -\frac{1}{6}x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\frac{1}{6}x^{3}+x-x+\frac{1}{4}x^{3})$$

Combine x and -x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\frac{1}{6}x^{3}+\frac{1}{4}x^{3})$$

Combine -\frac{1}{6}x^{3} and \frac{1}{4}x^{3} to get \frac{1}{12}x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{12}x^{3})$$

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

$$3\times \frac{1}{12}x^{3-1}$$

Multiply 3 times \frac{1}{12}.

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

Subtract 1 from 3.

$$\frac{1}{4}x^{2}$$