Solve 3n^2+n+4+7n^2+9n+5

$$3 n ^ { 2 } + n + 4 + 7 n ^ { 2 } + 9 n + 5$$

Evaluate

$$10n^{2}+10n+9$$

Solution Steps

Combine 3n^{2} and 7n^{2} to get 10n^{2}.

$$10n^{2}+n+4+9n+5$$

Combine n and 9n to get 10n.

$$10n^{2}+10n+4+5$$

Add 4 and 5 to get 9.

$$10n^{2}+10n+9$$

Differentiate w.r.t. n

$$20n+10$$

Steps Using Derivative Rule for Sum

Combine 3n^{2} and 7n^{2} to get 10n^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(10n^{2}+n+4+9n+5)$$

Combine n and 9n to get 10n.

$$\frac{\mathrm{d}}{\mathrm{d}n}(10n^{2}+10n+4+5)$$

Add 4 and 5 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}n}(10n^{2}+10n+9)$$

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}.

$$2\times 10n^{2-1}+10n^{1-1}$$

Multiply 2 times 10.

$$20n^{2-1}+10n^{1-1}$$

Subtract 1 from 2.

$$20n^{1}+10n^{1-1}$$

Subtract 1 from 1.

$$20n^{1}+10n^{0}$$

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

$$20n+10n^{0}$$

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

$$20n+10\times 1$$

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

$$20n+10$$