Solve a^4+2sqrt{2}a^3+2sqrt{2}a+1

$$a ^ { 4 } + 2 \sqrt { 2 } a ^ { 3 } + 2 \sqrt { 2 } a + 1$$

Differentiate w.r.t. a

$$4a^{3}+6\sqrt{2}a^{2}+2\sqrt{2}$$

Steps Using Derivative Rule for Sum

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$4a^{4-1}+3\times 2\sqrt{2}a^{3-1}+2\sqrt{2}a^{1-1}$$

Subtract 1 from 4.

$$4a^{3}+3\times 2\sqrt{2}a^{3-1}+2\sqrt{2}a^{1-1}$$

Multiply 3 times 2\sqrt{2}.

$$4a^{3}+6\sqrt{2}a^{3-1}+2\sqrt{2}a^{1-1}$$

Subtract 1 from 3.

$$4a^{3}+6\sqrt{2}a^{2}+2\sqrt{2}a^{1-1}$$

Subtract 1 from 1.

$$4a^{3}+6\sqrt{2}a^{2}+2\sqrt{2}a^{0}$$

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

$$4a^{3}+6\sqrt{2}a^{2}+2\sqrt{2}\times 1$$

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

$$4a^{3}+6\sqrt{2}a^{2}+2\sqrt{2}$$

Evaluate

$$a^{4}+2\sqrt{2}a^{3}+2\sqrt{2}a+1$$