Solve 1/2t^3*2/3t*1/8t^5=

$$\frac { 1 } { 2 } t ^ { 3 } \times \frac { 2 } { 3 } t \times \frac { 1 } { 8 } t ^ { 5 } =$$

Evaluate

$$\frac{t^{9}}{24}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{1}{2}t^{4}\times \frac{2}{3}\times \frac{1}{8}t^{5}$$

To multiply powers of the same base, add their exponents. Add 4 and 5 to get 9.

$$\frac{1}{2}t^{9}\times \frac{2}{3}\times \frac{1}{8}$$

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

$$\frac{1}{3}t^{9}\times \frac{1}{8}$$

Multiply \frac{1}{3} and \frac{1}{8} to get \frac{1}{24}.

$$\frac{1}{24}t^{9}$$

Differentiate w.r.t. t

$$\frac{3t^{8}}{8}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1}{2}t^{4}\times \frac{2}{3}\times \frac{1}{8}t^{5})$$

To multiply powers of the same base, add their exponents. Add 4 and 5 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1}{2}t^{9}\times \frac{2}{3}\times \frac{1}{8})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1}{3}t^{9}\times \frac{1}{8})$$

Multiply \frac{1}{3} and \frac{1}{8} to get \frac{1}{24}.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1}{24}t^{9})$$

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

$$9\times \frac{1}{24}t^{9-1}$$

Multiply 9 times \frac{1}{24}.

$$\frac{3}{8}t^{9-1}$$

Subtract 1 from 9.

$$\frac{3}{8}t^{8}$$