Solve 11/6a^{2}+2/3a-1/3ab^{3}-21/10a^2-frac{2}{3}a+frac{19}{15}a^2-frac{1}{6}ab^3+frac{1}{2}ab^3

$$\frac { 11 } { 6 } a ^ { 2 } + \frac { 2 } { 3 } a - \frac { 1 } { 3 } a b ^ { 3 } - \frac { 21 } { 10 } a ^ { 2 } - \frac { 2 } { 3 } a + \frac { 19 } { 15 } a ^ { 2 } - \frac { 1 } { 6 } a b ^ { 3 } + \frac { 1 } { 2 } a b ^ { 3 }$$

Evaluate

$$a^{2}$$

Solution Steps

Combine -\frac{21}{10}a^{2} and \frac{19}{15}a^{2} to get -\frac{5}{6}a^{2}.

$$\frac{11}{6}a^{2}+\frac{2}{3}a-\frac{1}{3}ab^{3}-\frac{5}{6}a^{2}-\frac{2}{3}a-\frac{1}{6}ab^{3}+\frac{1}{2}ab^{3}$$

Combine \frac{11}{6}a^{2} and -\frac{5}{6}a^{2} to get a^{2}.

$$a^{2}+\frac{2}{3}a-\frac{1}{3}ab^{3}-\frac{2}{3}a-\frac{1}{6}ab^{3}+\frac{1}{2}ab^{3}$$

Combine \frac{2}{3}a and -\frac{2}{3}a to get 0.

$$a^{2}-\frac{1}{3}ab^{3}-\frac{1}{6}ab^{3}+\frac{1}{2}ab^{3}$$

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

$$a^{2}-\frac{1}{2}ab^{3}+\frac{1}{2}ab^{3}$$

Combine -\frac{1}{2}ab^{3} and \frac{1}{2}ab^{3} to get 0.

$$a^{2}$$

Differentiate w.r.t. a

$$2a$$

Steps Using Definition of a Derivative

Combine -\frac{21}{10}a^{2} and \frac{19}{15}a^{2} to get -\frac{5}{6}a^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{11}{6}a^{2}+\frac{2}{3}a-\frac{1}{3}ab^{3}-\frac{5}{6}a^{2}-\frac{2}{3}a-\frac{1}{6}ab^{3}+\frac{1}{2}ab^{3})$$

Combine \frac{11}{6}a^{2} and -\frac{5}{6}a^{2} to get a^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{2}+\frac{2}{3}a-\frac{1}{3}ab^{3}-\frac{2}{3}a-\frac{1}{6}ab^{3}+\frac{1}{2}ab^{3})$$

Combine \frac{2}{3}a and -\frac{2}{3}a to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{2}-\frac{1}{3}ab^{3}-\frac{1}{6}ab^{3}+\frac{1}{2}ab^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{2}-\frac{1}{2}ab^{3}+\frac{1}{2}ab^{3})$$

Combine -\frac{1}{2}ab^{3} and \frac{1}{2}ab^{3} to get 0.

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

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

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

Subtract 1 from 2.

$$2a^{1}$$

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

$$2a$$