Solve (7x^3+8x^2-3x+6)+(3x^3-3x^2-15)

$$( 7 x ^ { 3 } + 8 x ^ { 2 } - 3 x + 6 ) + ( 3 x ^ { 3 } - 3 x ^ { 2 } - 15 )$$

Evaluate

$$10x^{3}+5x^{2}-3x-9$$

Solution Steps

Combine 7x^{3} and 3x^{3} to get 10x^{3}.

$$10x^{3}+8x^{2}-3x+6-3x^{2}-15$$

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

$$10x^{3}+5x^{2}-3x+6-15$$

Subtract 15 from 6 to get -9.

$$10x^{3}+5x^{2}-3x-9$$

Differentiate w.r.t. x

$$30x^{2}+10x-3$$

Steps Using Derivative Rule for Sum

Combine 7x^{3} and 3x^{3} to get 10x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{3}+8x^{2}-3x+6-3x^{2}-15)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{3}+5x^{2}-3x+6-15)$$

Subtract 15 from 6 to get -9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{3}+5x^{2}-3x-9)$$

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}.

$$3\times 10x^{3-1}+2\times 5x^{2-1}-3x^{1-1}$$

Multiply 3 times 10.

$$30x^{3-1}+2\times 5x^{2-1}-3x^{1-1}$$

Subtract 1 from 3.

$$30x^{2}+2\times 5x^{2-1}-3x^{1-1}$$

Multiply 2 times 5.

$$30x^{2}+10x^{2-1}-3x^{1-1}$$

Subtract 1 from 2.

$$30x^{2}+10x^{1}-3x^{1-1}$$

Subtract 1 from 1.

$$30x^{2}+10x^{1}-3x^{0}$$

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

$$30x^{2}+10x-3x^{0}$$

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

$$30x^{2}+10x-3$$