Solve 5a^2-6a+7a^2+3a-2+18a+7

$$5 a ^ { 2 } - 6 a + 7 a ^ { 2 } + 3 a - 2 + 18 a + 7$$

Evaluate

$$12a^{2}+15a+5$$

Solution Steps

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

$$12a^{2}-6a+3a-2+18a+7$$

Combine -6a and 3a to get -3a.

$$12a^{2}-3a-2+18a+7$$

Combine -3a and 18a to get 15a.

$$12a^{2}+15a-2+7$$

Add -2 and 7 to get 5.

$$12a^{2}+15a+5$$

Differentiate w.r.t. a

$$24a+15$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(12a^{2}-6a+3a-2+18a+7)$$

Combine -6a and 3a to get -3a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(12a^{2}-3a-2+18a+7)$$

Combine -3a and 18a to get 15a.

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

Add -2 and 7 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}a}(12a^{2}+15a+5)$$

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}.

$$2\times 12a^{2-1}+15a^{1-1}$$

Multiply 2 times 12.

$$24a^{2-1}+15a^{1-1}$$

Subtract 1 from 2.

$$24a^{1}+15a^{1-1}$$

Subtract 1 from 1.

$$24a^{1}+15a^{0}$$

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

$$24a+15a^{0}$$

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

$$24a+15\times 1$$

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

$$24a+15$$