Solve (5x^3+x+5+5x^2)+(2x+3x^2+4-4x^3)

$$( 5 x ^ { 3 } + x + 5 + 5 x ^ { 2 } ) + ( 2 x + 3 x ^ { 2 } + 4 - 4 x ^ { 3 } )$$

Evaluate

$$x^{3}+8x^{2}+3x+9$$

Solution Steps

Combine x and 2x to get 3x.

$$5x^{3}+3x+5+5x^{2}+3x^{2}+4-4x^{3}$$

Combine 5x^{2} and 3x^{2} to get 8x^{2}.

$$5x^{3}+3x+5+8x^{2}+4-4x^{3}$$

Add 5 and 4 to get 9.

$$5x^{3}+3x+9+8x^{2}-4x^{3}$$

Combine 5x^{3} and -4x^{3} to get x^{3}.

$$x^{3}+3x+9+8x^{2}$$

Differentiate w.r.t. x

$$3x^{2}+16x+3$$

Steps Using Derivative Rule for Sum

Combine x and 2x to get 3x.

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

Combine 5x^{2} and 3x^{2} to get 8x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}+3x+5+8x^{2}+4-4x^{3})$$

Add 5 and 4 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}+3x+9+8x^{2}-4x^{3})$$

Combine 5x^{3} and -4x^{3} to get x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}+3x+9+8x^{2})$$

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}.

$$3x^{3-1}+3x^{1-1}+2\times 8x^{2-1}$$

Subtract 1 from 3.

$$3x^{2}+3x^{1-1}+2\times 8x^{2-1}$$

Subtract 1 from 1.

$$3x^{2}+3x^{0}+2\times 8x^{2-1}$$

Multiply 1 times 3.

$$3x^{2}+3x^{0}+16x^{2-1}$$

Subtract 1 from 2.

$$3x^{2}+3x^{0}+16x^{1}$$

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

$$3x^{2}+3x^{0}+16x$$

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

$$3x^{2}+3\times 1+16x$$

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

$$3x^{2}+3+16x$$