Solve -5x^4-16x^3+63x^2-16*-32

$$-5 { x }^{ 4 } -16 { x }^{ 3 } +63 { x }^{ 2 } -16 \times -32$$

Evaluate

$$512+63x^{2}-16x^{3}-5x^{4}$$

Solution Steps

Multiply 16 and -32 to get -512.

$$-5x^{4}-16x^{3}+63x^{2}-\left(-512\right)$$

The opposite of -512 is 512.

$$-5x^{4}-16x^{3}+63x^{2}+512$$

Differentiate w.r.t. x

$$2x\left(63-24x-10x^{2}\right)$$

Steps Using Derivative Rule for Sum

Multiply 16 and -32 to get -512.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-5x^{4}-16x^{3}+63x^{2}-\left(-512\right))$$

The opposite of -512 is 512.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-5x^{4}-16x^{3}+63x^{2}+512)$$

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}.

$$4\left(-5\right)x^{4-1}+3\left(-16\right)x^{3-1}+2\times 63x^{2-1}$$

Multiply 4 times -5.

$$-20x^{4-1}+3\left(-16\right)x^{3-1}+2\times 63x^{2-1}$$

Subtract 1 from 4.

$$-20x^{3}+3\left(-16\right)x^{3-1}+2\times 63x^{2-1}$$

Multiply 3 times -16.

$$-20x^{3}-48x^{3-1}+2\times 63x^{2-1}$$

Subtract 1 from 3.

$$-20x^{3}-48x^{2}+2\times 63x^{2-1}$$

Multiply 3 times -16.

$$-20x^{3}-48x^{2}+126x^{2-1}$$

Subtract 1 from 2.

$$-20x^{3}-48x^{2}+126x^{1}$$

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

$$-20x^{3}-48x^{2}+126x$$