Solve (-5/2b)(-b^{2})+(-4/3b^{2})(5/2b)+3/2b(b^2)+(-frac{1}{3}b^2)(frac{3}{2}b)+(-frac{1}{2}b^2)(frac{5}{2}b)+b^3

$$( - \frac { 5 } { 2 } b ) ( - b ^ { 2 } ) + ( - \frac { 4 } { 3 } b ^ { 2 } ) ( \frac { 5 } { 2 } b ) + \frac { 3 } { 2 } b ( b ^ { 2 } ) + ( - \frac { 1 } { 3 } b ^ { 2 } ) ( \frac { 3 } { 2 } b ) + ( - \frac { 1 } { 2 } b ^ { 2 } ) ( \frac { 5 } { 2 } b ) + b ^ { 3 }$$

Evaluate

$$-\frac{b^{3}}{12}$$

Solution Steps

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

$$-\frac{5}{2}b\left(-b^{2}\right)-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}bb^{2}-\frac{1}{3}b^{2}\times \frac{3}{2}b-\frac{1}{2}b^{2}\times \frac{5}{2}b+b^{3}$$

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

$$-\frac{5}{2}b\left(-b^{2}\right)-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{2}\times \frac{3}{2}b-\frac{1}{2}b^{2}\times \frac{5}{2}b+b^{3}$$

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

$$-\frac{5}{2}b\left(-b^{2}\right)-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{2}\times \frac{5}{2}b+b^{3}$$

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

$$-\frac{5}{2}b\left(-b^{2}\right)-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3}$$

Multiply -\frac{5}{2} and -1 to get \frac{5}{2}.

$$\frac{5}{2}bb^{2}-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3}$$

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

$$\frac{5}{2}b^{3}-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3}$$

Multiply -\frac{4}{3} and \frac{5}{2} to get -\frac{10}{3}.

$$\frac{5}{2}b^{3}-\frac{10}{3}b^{3}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3}$$

Combine \frac{5}{2}b^{3} and -\frac{10}{3}b^{3} to get -\frac{5}{6}b^{3}.

$$-\frac{5}{6}b^{3}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3}$$

Combine -\frac{5}{6}b^{3} and \frac{3}{2}b^{3} to get \frac{2}{3}b^{3}.

$$\frac{2}{3}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3}$$

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

$$\frac{2}{3}b^{3}-\frac{1}{2}b^{3}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3}$$

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

$$\frac{1}{6}b^{3}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3}$$

Multiply -\frac{1}{2} and \frac{5}{2} to get -\frac{5}{4}.

$$\frac{1}{6}b^{3}-\frac{5}{4}b^{3}+b^{3}$$

Combine \frac{1}{6}b^{3} and -\frac{5}{4}b^{3} to get -\frac{13}{12}b^{3}.

$$-\frac{13}{12}b^{3}+b^{3}$$

Combine -\frac{13}{12}b^{3} and b^{3} to get -\frac{1}{12}b^{3}.

$$-\frac{1}{12}b^{3}$$

Differentiate w.r.t. b

$$-\frac{b^{2}}{4}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(-\frac{5}{2}b\left(-b^{2}\right)-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}bb^{2}-\frac{1}{3}b^{2}\times \frac{3}{2}b-\frac{1}{2}b^{2}\times \frac{5}{2}b+b^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(-\frac{5}{2}b\left(-b^{2}\right)-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{2}\times \frac{3}{2}b-\frac{1}{2}b^{2}\times \frac{5}{2}b+b^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(-\frac{5}{2}b\left(-b^{2}\right)-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{2}\times \frac{5}{2}b+b^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(-\frac{5}{2}b\left(-b^{2}\right)-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3})$$

Multiply -\frac{5}{2} and -1 to get \frac{5}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{5}{2}bb^{2}-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{5}{2}b^{3}-\frac{4}{3}b^{3}\times \frac{5}{2}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3})$$

Multiply -\frac{4}{3} and \frac{5}{2} to get -\frac{10}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{5}{2}b^{3}-\frac{10}{3}b^{3}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3})$$

Combine \frac{5}{2}b^{3} and -\frac{10}{3}b^{3} to get -\frac{5}{6}b^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(-\frac{5}{6}b^{3}+\frac{3}{2}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3})$$

Combine -\frac{5}{6}b^{3} and \frac{3}{2}b^{3} to get \frac{2}{3}b^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{2}{3}b^{3}-\frac{1}{3}b^{3}\times \frac{3}{2}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{2}{3}b^{3}-\frac{1}{2}b^{3}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{6}b^{3}-\frac{1}{2}b^{3}\times \frac{5}{2}+b^{3})$$

Multiply -\frac{1}{2} and \frac{5}{2} to get -\frac{5}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{6}b^{3}-\frac{5}{4}b^{3}+b^{3})$$

Combine \frac{1}{6}b^{3} and -\frac{5}{4}b^{3} to get -\frac{13}{12}b^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(-\frac{13}{12}b^{3}+b^{3})$$

Combine -\frac{13}{12}b^{3} and b^{3} to get -\frac{1}{12}b^{3}.

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

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

$$3\left(-\frac{1}{12}\right)b^{3-1}$$

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

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

Subtract 1 from 3.

$$-\frac{1}{4}b^{2}$$