Solve 3/20a+11/30a-1/12a+1/15a)-(frac{3}{8}a-frac{7}{20}a-frac{2}{5}a)

$$\frac { 3 } { 20 } a + \frac { 11 } { 30 } a - \frac { 1 } { 12 } a + \frac { 1 } { 15 } a ) - ( \frac { 3 } { 8 } a - \frac { 7 } { 20 } a - \frac { 2 } { 5 } a )$$

Evaluate

$$\frac{a}{2}$$

Solution Steps

Combine \frac{3}{20}a and \frac{11}{30}a to get \frac{31}{60}a.

$$\frac{31}{60}a-\frac{1}{12}a+\frac{1}{15}a$$

Combine \frac{31}{60}a and -\frac{1}{12}a to get \frac{13}{30}a.

$$\frac{13}{30}a+\frac{1}{15}a$$

Combine \frac{13}{30}a and \frac{1}{15}a to get \frac{1}{2}a.

$$\frac{1}{2}a$$

Differentiate w.r.t. a

$$\frac{1}{2} = 0.5$$

Steps Using Definition of a Derivative

Combine \frac{3}{20}a and \frac{11}{30}a to get \frac{31}{60}a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{31}{60}a-\frac{1}{12}a+\frac{1}{15}a)$$

Combine \frac{31}{60}a and -\frac{1}{12}a to get \frac{13}{30}a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{13}{30}a+\frac{1}{15}a)$$

Combine \frac{13}{30}a and \frac{1}{15}a to get \frac{1}{2}a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{2}a)$$

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

$$\frac{1}{2}a^{1-1}$$

Subtract 1 from 1.

$$\frac{1}{2}a^{0}$$

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

$$\frac{1}{2}\times 1$$

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

$$\frac{1}{2}$$