Solve x^{6}+0x^5+2x^4+0x^3-x^2+0x+0divx+1

$$x ^ { 6 } + 0 x ^ { 5 } + 2 x ^ { 4 } + 0 x ^ { 3 } - x ^ { 2 } + 0 x + 0 \div x + 1$$

Evaluate

$$x^{6}+2x^{4}-x^{2}+1$$

Solution Steps

Anything times zero gives zero.

$$x^{6}+0+2x^{4}+0x^{3}-x^{2}+0x+\frac{0}{x}+1$$

Anything times zero gives zero.

$$x^{6}+0+2x^{4}+0-x^{2}+0x+\frac{0}{x}+1$$

Add 0 and 0 to get 0.

$$x^{6}+2x^{4}-x^{2}+0x+\frac{0}{x}+1$$

Anything times zero gives zero.

$$x^{6}+2x^{4}-x^{2}+0+\frac{0}{x}+1$$

Zero divided by any non-zero term gives zero.

$$x^{6}+2x^{4}-x^{2}+0+0+1$$

Add 0 and 0 to get 0.

$$x^{6}+2x^{4}-x^{2}+1$$

Differentiate w.r.t. x

$$6x^{5}+8x^{3}-2x$$

Steps Using Derivative Rule for Sum

Anything times zero gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+0+2x^{4}+0x^{3}-x^{2}+0x+\frac{0}{x}+1)$$

Anything times zero gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+0+2x^{4}+0-x^{2}+0x+\frac{0}{x}+1)$$

Add 0 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+2x^{4}-x^{2}+0x+\frac{0}{x}+1)$$

Anything times zero gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+2x^{4}-x^{2}+0+\frac{0}{x}+1)$$

Zero divided by any non-zero term gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+2x^{4}-x^{2}+0+0+1)$$

Add 0 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+2x^{4}-x^{2}+1)$$

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}.

$$6x^{6-1}+4\times 2x^{4-1}+2\left(-1\right)x^{2-1}$$

Subtract 1 from 6.

$$6x^{5}+4\times 2x^{4-1}+2\left(-1\right)x^{2-1}$$

Multiply 4 times 2.

$$6x^{5}+8x^{4-1}+2\left(-1\right)x^{2-1}$$

Subtract 1 from 4.

$$6x^{5}+8x^{3}+2\left(-1\right)x^{2-1}$$

Multiply 4 times 2.

$$6x^{5}+8x^{3}-2x^{2-1}$$

Subtract 1 from 2.

$$6x^{5}+8x^{3}-2x^{1}$$

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

$$6x^{5}+8x^{3}-2x$$