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

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

Evaluate

$$6x^{3}-5x^{2}+4x+3$$

Solution Steps

Combine 5x^{3} and x^{3} to get 6x^{3}.

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

Combine -x^{2} and -4x^{2} to get -5x^{2}.

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

Combine 3x and x to get 4x.

$$6x^{3}-5x^{2}+4x+3$$

Differentiate w.r.t. x

$$18x^{2}-10x+4$$

Steps Using Derivative Rule for Sum

Combine 5x^{3} and x^{3} to get 6x^{3}.

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

Combine -x^{2} and -4x^{2} to get -5x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}-5x^{2}+3x+3+x)$$

Combine 3x and x to get 4x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}-5x^{2}+4x+3)$$

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}.

$$3\times 6x^{3-1}+2\left(-5\right)x^{2-1}+4x^{1-1}$$

Multiply 3 times 6.

$$18x^{3-1}+2\left(-5\right)x^{2-1}+4x^{1-1}$$

Subtract 1 from 3.

$$18x^{2}+2\left(-5\right)x^{2-1}+4x^{1-1}$$

Multiply 2 times -5.

$$18x^{2}-10x^{2-1}+4x^{1-1}$$

Subtract 1 from 2.

$$18x^{2}-10x^{1}+4x^{1-1}$$

Subtract 1 from 1.

$$18x^{2}-10x^{1}+4x^{0}$$

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

$$18x^{2}-10x+4x^{0}$$

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

$$18x^{2}-10x+4\times 1$$

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

$$18x^{2}-10x+4$$