Solve (1/2x2/3)+(2/3x5/2)

$$( \frac { 1 } { 2 } x \frac { 2 } { 3 } ) + ( \frac { 2 } { 3 } x \frac { 5 } { 2 } )$$

Evaluate

$$2x$$

Solution Steps

Multiply \frac{1}{2} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2}{2\times 3}x+\frac{2}{3}x\times \frac{5}{2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{1}{3}x+\frac{2}{3}x\times \frac{5}{2}$$

Multiply \frac{2}{3} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{3}x+\frac{2\times 5}{3\times 2}x$$

Cancel out 2 in both numerator and denominator.

$$\frac{1}{3}x+\frac{5}{3}x$$

Combine \frac{1}{3}x and \frac{5}{3}x to get 2x.

$$2x$$

Differentiate w.r.t. x

$$2$$

Steps Using Definition of a Derivative

Multiply \frac{1}{2} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1\times 2}{2\times 3}x+\frac{2}{3}x\times \frac{5}{2})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3}x+\frac{2}{3}x\times \frac{5}{2})$$

Multiply \frac{2}{3} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3}x+\frac{2\times 5}{3\times 2}x)$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3}x+\frac{5}{3}x)$$

Combine \frac{1}{3}x and \frac{5}{3}x to get 2x.

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

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

$$2x^{1-1}$$

Subtract 1 from 1.

$$2x^{0}$$

For any term t except 0, t^{0}=1.

$$2\times 1$$

For any term t, t\times 1=t and 1t=t.

$$2$$