Solve (5c^2+7c+1)+(8c^3+4c^2+3c+1)

$$( 5 c ^ { 2 } + 7 c + 1 ) + ( 8 c ^ { 3 } + 4 c ^ { 2 } + 3 c + 1 )$$

Evaluate

$$8c^{3}+9c^{2}+10c+2$$

Solution Steps

Combine 5c^{2} and 4c^{2} to get 9c^{2}.

$$9c^{2}+7c+1+8c^{3}+3c+1$$

Combine 7c and 3c to get 10c.

$$9c^{2}+10c+1+8c^{3}+1$$

Add 1 and 1 to get 2.

$$9c^{2}+10c+2+8c^{3}$$

Differentiate w.r.t. c

$$24c^{2}+18c+10$$

Steps Using Derivative Rule for Sum

Combine 5c^{2} and 4c^{2} to get 9c^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(9c^{2}+7c+1+8c^{3}+3c+1)$$

Combine 7c and 3c to get 10c.

$$\frac{\mathrm{d}}{\mathrm{d}c}(9c^{2}+10c+1+8c^{3}+1)$$

Add 1 and 1 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}c}(9c^{2}+10c+2+8c^{3})$$

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}.

$$2\times 9c^{2-1}+10c^{1-1}+3\times 8c^{3-1}$$

Multiply 2 times 9.

$$18c^{2-1}+10c^{1-1}+3\times 8c^{3-1}$$

Subtract 1 from 2.

$$18c^{1}+10c^{1-1}+3\times 8c^{3-1}$$

Subtract 1 from 1.

$$18c^{1}+10c^{0}+3\times 8c^{3-1}$$

Multiply 1 times 10.

$$18c^{1}+10c^{0}+24c^{3-1}$$

Subtract 1 from 3.

$$18c^{1}+10c^{0}+24c^{2}$$

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

$$18c+10c^{0}+24c^{2}$$

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

$$18c+10\times 1+24c^{2}$$

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

$$18c+10+24c^{2}$$