Solve 7x^3+2x-5x^2+6+x+9+3x^3+12x^2=

$$7 { x }^{ 3 } +2x-5 { x }^{ 2 } +6+x+9+3 { x }^{ 3 } +12 { x }^{ 2 } =$$

Evaluate

$$10x^{3}+7x^{2}+3x+15$$

Solution Steps

Combine 2x and x to get 3x.

$$7x^{3}+3x-5x^{2}+6+9+3x^{3}+12x^{2}$$

Add 6 and 9 to get 15.

$$7x^{3}+3x-5x^{2}+15+3x^{3}+12x^{2}$$

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

$$10x^{3}+3x-5x^{2}+15+12x^{2}$$

Combine -5x^{2} and 12x^{2} to get 7x^{2}.

$$10x^{3}+3x+7x^{2}+15$$

Differentiate w.r.t. x

$$30x^{2}+14x+3$$

Steps Using Derivative Rule for Sum

Combine 2x and x to get 3x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7x^{3}+3x-5x^{2}+6+9+3x^{3}+12x^{2})$$

Add 6 and 9 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7x^{3}+3x-5x^{2}+15+3x^{3}+12x^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{3}+3x-5x^{2}+15+12x^{2})$$

Combine -5x^{2} and 12x^{2} to get 7x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{3}+3x+7x^{2}+15)$$

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}.

$$3\times 10x^{3-1}+3x^{1-1}+2\times 7x^{2-1}$$

Multiply 3 times 10.

$$30x^{3-1}+3x^{1-1}+2\times 7x^{2-1}$$

Subtract 1 from 3.

$$30x^{2}+3x^{1-1}+2\times 7x^{2-1}$$

Subtract 1 from 1.

$$30x^{2}+3x^{0}+2\times 7x^{2-1}$$

Multiply 1 times 3.

$$30x^{2}+3x^{0}+14x^{2-1}$$

Subtract 1 from 2.

$$30x^{2}+3x^{0}+14x^{1}$$

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

$$30x^{2}+3x^{0}+14x$$

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

$$30x^{2}+3\times 1+14x$$

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

$$30x^{2}+3+14x$$