Solve 8+x-4x^2sqrt{6+x^2+7x^4}

$$8+x-4 { x }^{ 2 } \sqrt{ 6+ { x }^{ 2 } +7 { x }^{ 4 } }$$

Differentiate w.r.t. x

$$\frac{-8x\left(7x^{4}+x^{2}+6\right)-4\left(14x^{2}+1\right)x^{3}+\sqrt{7x^{4}+x^{2}+6}}{\sqrt{7x^{4}+x^{2}+6}}$$

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}.

$$x^{1-1}+2\left(-4\sqrt{7x^{4}+x^{2}+6}\right)x^{2-1}$$

Subtract 1 from 1.

$$x^{0}+2\left(-4\sqrt{7x^{4}+x^{2}+6}\right)x^{2-1}$$

Multiply 2 times -4\sqrt{6+x^{2}+7x^{4}}.

$$x^{0}+\left(-8\sqrt{7x^{4}+x^{2}+6}\right)x^{2-1}$$

Subtract 1 from 2.

$$x^{0}+\left(-8\sqrt{7x^{4}+x^{2}+6}\right)x^{1}$$

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

$$x^{0}+\left(-8\sqrt{7x^{4}+x^{2}+6}\right)x$$

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

$$1+\left(-8\sqrt{7x^{4}+x^{2}+6}\right)x$$

Evaluate

$$-4x^{2}\sqrt{7x^{4}+x^{2}+6}+x+8$$