Solve 16x^4+8x^2+7x-5quad2x^2-x+5

$$16 x ^ { 4 } + 8 x ^ { 2 } + 7 x - 5 \quad 2 x ^ { 2 } - x + 5$$

Evaluate

$$16x^{4}-2x^{2}+6x+5$$

Solution Steps

Multiply 5 and 2 to get 10.

$$16x^{4}+8x^{2}+7x-10x^{2}-x+5$$

Combine 8x^{2} and -10x^{2} to get -2x^{2}.

$$16x^{4}-2x^{2}+7x-x+5$$

Combine 7x and -x to get 6x.

$$16x^{4}-2x^{2}+6x+5$$

Differentiate w.r.t. x

$$64x^{3}-4x+6$$

Steps Using Derivative Rule for Sum

Multiply 5 and 2 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16x^{4}+8x^{2}+7x-10x^{2}-x+5)$$

Combine 8x^{2} and -10x^{2} to get -2x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16x^{4}-2x^{2}+7x-x+5)$$

Combine 7x and -x to get 6x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16x^{4}-2x^{2}+6x+5)$$

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}.

$$4\times 16x^{4-1}+2\left(-2\right)x^{2-1}+6x^{1-1}$$

Multiply 4 times 16.

$$64x^{4-1}+2\left(-2\right)x^{2-1}+6x^{1-1}$$

Subtract 1 from 4.

$$64x^{3}+2\left(-2\right)x^{2-1}+6x^{1-1}$$

Multiply 2 times -2.

$$64x^{3}-4x^{2-1}+6x^{1-1}$$

Subtract 1 from 2.

$$64x^{3}-4x^{1}+6x^{1-1}$$

Subtract 1 from 1.

$$64x^{3}-4x^{1}+6x^{0}$$

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

$$64x^{3}-4x+6x^{0}$$

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

$$64x^{3}-4x+6\times 1$$

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

$$64x^{3}-4x+6$$