Solve (3x^3+5x^2-1)+(x^3-4x^2+2x-5)

$$( 3 x ^ { 3 } + 5 x ^ { 2 } - 1 ) + ( x ^ { 3 } - 4 x ^ { 2 } + 2 x - 5 )$$

Evaluate

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

Solution Steps

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

$$4x^{3}+5x^{2}-1-4x^{2}+2x-5$$

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

$$4x^{3}+x^{2}-1+2x-5$$

Subtract 5 from -1 to get -6.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{3}+5x^{2}-1-4x^{2}+2x-5)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{3}+x^{2}-1+2x-5)$$

Subtract 5 from -1 to get -6.

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

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}.

$$3\times 4x^{3-1}+2x^{2-1}+2x^{1-1}$$

Multiply 3 times 4.

$$12x^{3-1}+2x^{2-1}+2x^{1-1}$$

Subtract 1 from 3.

$$12x^{2}+2x^{2-1}+2x^{1-1}$$

Subtract 1 from 2.

$$12x^{2}+2x^{1}+2x^{1-1}$$

Subtract 1 from 1.

$$12x^{2}+2x^{1}+2x^{0}$$

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

$$12x^{2}+2x+2x^{0}$$

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

$$12x^{2}+2x+2\times 1$$

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

$$12x^{2}+2x+2$$