Solve (4x^3-3x^2+7x-2)+(x^3+7x^2-4x+5)

$$( 4 x ^ { 3 } - 3 x ^ { 2 } + 7 x - 2 ) + ( x ^ { 3 } + 7 x ^ { 2 } - 4 x + 5 )$$

Evaluate

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

Solution Steps

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

$$5x^{3}-3x^{2}+7x-2+7x^{2}-4x+5$$

Combine -3x^{2} and 7x^{2} to get 4x^{2}.

$$5x^{3}+4x^{2}+7x-2-4x+5$$

Combine 7x and -4x to get 3x.

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

Add -2 and 5 to get 3.

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

Differentiate w.r.t. x

$$15x^{2}+8x+3$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}-3x^{2}+7x-2+7x^{2}-4x+5)$$

Combine -3x^{2} and 7x^{2} to get 4x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}+4x^{2}+7x-2-4x+5)$$

Combine 7x and -4x to get 3x.

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

Add -2 and 5 to get 3.

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

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}.

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

Multiply 3 times 5.

$$15x^{3-1}+2\times 4x^{2-1}+3x^{1-1}$$

Subtract 1 from 3.

$$15x^{2}+2\times 4x^{2-1}+3x^{1-1}$$

Multiply 2 times 4.

$$15x^{2}+8x^{2-1}+3x^{1-1}$$

Subtract 1 from 2.

$$15x^{2}+8x^{1}+3x^{1-1}$$

Subtract 1 from 1.

$$15x^{2}+8x^{1}+3x^{0}$$

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

$$15x^{2}+8x+3x^{0}$$

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

$$15x^{2}+8x+3\times 1$$

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

$$15x^{2}+8x+3$$