Solve 6.4*6.4*6.4*6.4*b*b*b

$$6.4 \cdot 6.4 \cdot 6.4 \cdot 6.4 \cdot b \cdot b \cdot b$$

Evaluate

$$\frac{1048576b^{3}}{625}$$

Solution Steps

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

$$6.4\times 6.4\times 6.4\times 6.4b^{2}b$$

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

$$6.4\times 6.4\times 6.4\times 6.4b^{3}$$

Multiply 6.4 and 6.4 to get 40.96.

$$40.96\times 6.4\times 6.4b^{3}$$

Multiply 40.96 and 6.4 to get 262.144.

$$262.144\times 6.4b^{3}$$

Multiply 262.144 and 6.4 to get 1677.7216.

$$1677.7216b^{3}$$

Differentiate w.r.t. b

$$\frac{3145728b^{2}}{625}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(6.4\times 6.4\times 6.4\times 6.4b^{2}b)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(6.4\times 6.4\times 6.4\times 6.4b^{3})$$

Multiply 6.4 and 6.4 to get 40.96.

$$\frac{\mathrm{d}}{\mathrm{d}b}(40.96\times 6.4\times 6.4b^{3})$$

Multiply 40.96 and 6.4 to get 262.144.

$$\frac{\mathrm{d}}{\mathrm{d}b}(262.144\times 6.4b^{3})$$

Multiply 262.144 and 6.4 to get 1677.7216.

$$\frac{\mathrm{d}}{\mathrm{d}b}(1677.7216b^{3})$$

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

$$3\times 1677.7216b^{3-1}$$

Multiply 3 times 1677.7216.

$$5033.1648b^{3-1}$$

Subtract 1 from 3.

$$5033.1648b^{2}$$