Solve (4v^2-5v+3)+(-v^2+6v+7)

$$( 4 v ^ { 2 } - 5 v + 3 ) + ( - v ^ { 2 } + 6 v + 7 )$$

Evaluate

$$3v^{2}+v+10$$

Solution Steps

Combine -5v and 6v to get v.

$$4v^{2}+v+3-v^{2}+7$$

Add 3 and 7 to get 10.

$$4v^{2}+v+10-v^{2}$$

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

$$3v^{2}+v+10$$

Differentiate w.r.t. v

$$6v+1$$

Steps Using Derivative Rule for Sum

Combine -5v and 6v to get v.

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

Add 3 and 7 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}v}(4v^{2}+v+10-v^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}v}(3v^{2}+v+10)$$

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}.

$$2\times 3v^{2-1}+v^{1-1}$$

Multiply 2 times 3.

$$6v^{2-1}+v^{1-1}$$

Subtract 1 from 2.

$$6v^{1}+v^{1-1}$$

Subtract 1 from 1.

$$6v^{1}+v^{0}$$

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

$$6v+v^{0}$$

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

$$6v+1$$