Solve (7y^3+y^2+6y+8)+(6y^3+5y^2+y+7)

$$( 7 y ^ { 3 } + y ^ { 2 } + 6 y + 8 ) + ( 6 y ^ { 3 } + 5 y ^ { 2 } + y + 7 )$$

Evaluate

$$13y^{3}+6y^{2}+7y+15$$

Solution Steps

Combine 7y^{3} and 6y^{3} to get 13y^{3}.

$$13y^{3}+y^{2}+6y+8+5y^{2}+y+7$$

Combine y^{2} and 5y^{2} to get 6y^{2}.

$$13y^{3}+6y^{2}+6y+8+y+7$$

Combine 6y and y to get 7y.

$$13y^{3}+6y^{2}+7y+8+7$$

Add 8 and 7 to get 15.

$$13y^{3}+6y^{2}+7y+15$$

Differentiate w.r.t. y

$$39y^{2}+12y+7$$

Steps Using Derivative Rule for Sum

Combine 7y^{3} and 6y^{3} to get 13y^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(13y^{3}+y^{2}+6y+8+5y^{2}+y+7)$$

Combine y^{2} and 5y^{2} to get 6y^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(13y^{3}+6y^{2}+6y+8+y+7)$$

Combine 6y and y to get 7y.

$$\frac{\mathrm{d}}{\mathrm{d}y}(13y^{3}+6y^{2}+7y+8+7)$$

Add 8 and 7 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}y}(13y^{3}+6y^{2}+7y+15)$$

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}.

$$3\times 13y^{3-1}+2\times 6y^{2-1}+7y^{1-1}$$

Multiply 3 times 13.

$$39y^{3-1}+2\times 6y^{2-1}+7y^{1-1}$$

Subtract 1 from 3.

$$39y^{2}+2\times 6y^{2-1}+7y^{1-1}$$

Multiply 2 times 6.

$$39y^{2}+12y^{2-1}+7y^{1-1}$$

Subtract 1 from 2.

$$39y^{2}+12y^{1}+7y^{1-1}$$

Subtract 1 from 1.

$$39y^{2}+12y^{1}+7y^{0}$$

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

$$39y^{2}+12y+7y^{0}$$

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

$$39y^{2}+12y+7\times 1$$

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

$$39y^{2}+12y+7$$