Solve 5sqrt{8}quadtext{(b)}2sqrt{2}

$$5 \sqrt { 8 } \quad \text { (b) } 2 \sqrt { 2 }$$

Evaluate

$$40b$$

Solution Steps

Factor 8=2\times 4. Rewrite the square root of the product \sqrt{2\times 4} as the product of square roots \sqrt{2}\sqrt{4}.

$$2\times 5\sqrt{2}\sqrt{4}b\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$2\times 5\times 2b\sqrt{4}$$

Multiply 2 and 5 to get 10.

$$10\times 2b\sqrt{4}$$

Multiply 10 and 2 to get 20.

$$20b\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$20b\times 2$$

Multiply 20 and 2 to get 40.

$$40b$$

Differentiate w.r.t. b

$$40$$

Steps Using Definition of a Derivative

Factor 8=2\times 4. Rewrite the square root of the product \sqrt{2\times 4} as the product of square roots \sqrt{2}\sqrt{4}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(2\times 5\sqrt{2}\sqrt{4}b\sqrt{2})$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}b}(2\times 5\times 2b\sqrt{4})$$

Multiply 2 and 5 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}b}(10\times 2b\sqrt{4})$$

Multiply 10 and 2 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}b}(20b\sqrt{4})$$

Calculate the square root of 4 and get 2.

$$\frac{\mathrm{d}}{\mathrm{d}b}(20b\times 2)$$

Multiply 20 and 2 to get 40.

$$\frac{\mathrm{d}}{\mathrm{d}b}(40b)$$

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

$$40b^{1-1}$$

Subtract 1 from 1.

$$40b^{0}$$

For any term t except 0, t^{0}=1.

$$40\times 1$$

For any term t, t\times 1=t and 1t=t.

$$40$$