Solve -7/3a(-a^2)-a(-1/3a)(2a)+2/5a(-5a^2)+a^3

$$- \frac { 7 } { 3 } a ( - a ^ { 2 } ) - a ( - \frac { 1 } { 3 } a ) ( 2 a ) + \frac { 2 } { 5 } a ( - 5 a ^ { 2 } ) + a ^ { 3 }$$

Evaluate

$$2a^{3}$$

Solution Steps

Multiply a and a to get a^{2}.

$$-\frac{7}{3}a\left(-a^{2}\right)-a^{2}\left(-\frac{1}{3}\right)\times 2a+\frac{2}{5}a\left(-5\right)a^{2}+a^{3}$$

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

$$-\frac{7}{3}a\left(-a^{2}\right)-a^{3}\left(-\frac{1}{3}\right)\times 2+\frac{2}{5}a\left(-5\right)a^{2}+a^{3}$$

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

$$-\frac{7}{3}a\left(-a^{2}\right)-a^{3}\left(-\frac{1}{3}\right)\times 2+\frac{2}{5}a^{3}\left(-5\right)+a^{3}$$

Multiply -\frac{7}{3} and -1 to get \frac{7}{3}.

$$\frac{7}{3}aa^{2}-a^{3}\left(-\frac{1}{3}\right)\times 2+\frac{2}{5}a^{3}\left(-5\right)+a^{3}$$

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

$$\frac{7}{3}a^{3}-a^{3}\left(-\frac{1}{3}\right)\times 2+\frac{2}{5}a^{3}\left(-5\right)+a^{3}$$

Multiply -\frac{1}{3} and 2 to get -\frac{2}{3}.

$$\frac{7}{3}a^{3}-a^{3}\left(-\frac{2}{3}\right)+\frac{2}{5}a^{3}\left(-5\right)+a^{3}$$

Combine \frac{7}{3}a^{3} and -a^{3}\left(-\frac{2}{3}\right) to get 3a^{3}.

$$3a^{3}+\frac{2}{5}a^{3}\left(-5\right)+a^{3}$$

Multiply \frac{2}{5} and -5 to get -2.

$$3a^{3}-2a^{3}+a^{3}$$

Combine 3a^{3} and -2a^{3} to get a^{3}.

$$a^{3}+a^{3}$$

Combine a^{3} and a^{3} to get 2a^{3}.

$$2a^{3}$$

Differentiate w.r.t. a

$$6a^{2}$$

Steps Using Definition of a Derivative

Multiply a and a to get a^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-\frac{7}{3}a\left(-a^{2}\right)-a^{2}\left(-\frac{1}{3}\right)\times 2a+\frac{2}{5}a\left(-5\right)a^{2}+a^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(-\frac{7}{3}a\left(-a^{2}\right)-a^{3}\left(-\frac{1}{3}\right)\times 2+\frac{2}{5}a\left(-5\right)a^{2}+a^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(-\frac{7}{3}a\left(-a^{2}\right)-a^{3}\left(-\frac{1}{3}\right)\times 2+\frac{2}{5}a^{3}\left(-5\right)+a^{3})$$

Multiply -\frac{7}{3} and -1 to get \frac{7}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{7}{3}aa^{2}-a^{3}\left(-\frac{1}{3}\right)\times 2+\frac{2}{5}a^{3}\left(-5\right)+a^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{7}{3}a^{3}-a^{3}\left(-\frac{1}{3}\right)\times 2+\frac{2}{5}a^{3}\left(-5\right)+a^{3})$$

Multiply -\frac{1}{3} and 2 to get -\frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{7}{3}a^{3}-a^{3}\left(-\frac{2}{3}\right)+\frac{2}{5}a^{3}\left(-5\right)+a^{3})$$

Combine \frac{7}{3}a^{3} and -a^{3}\left(-\frac{2}{3}\right) to get 3a^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{3}+\frac{2}{5}a^{3}\left(-5\right)+a^{3})$$

Multiply \frac{2}{5} and -5 to get -2.

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

Combine 3a^{3} and -2a^{3} to get a^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{3}+a^{3})$$

Combine a^{3} and a^{3} to get 2a^{3}.

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

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

$$3\times 2a^{3-1}$$

Multiply 3 times 2.

$$6a^{3-1}$$

Subtract 1 from 3.

$$6a^{2}$$