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

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

Evaluate

$$8a^{4}+10a^{2}-a+7$$

Solution Steps

Add 3 and 4 to get 7.

$$6a+3a^{2}+7+3a^{4}+7a^{2}+5a^{4}-7a$$

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

$$6a+10a^{2}+7+3a^{4}+5a^{4}-7a$$

Combine 3a^{4} and 5a^{4} to get 8a^{4}.

$$6a+10a^{2}+7+8a^{4}-7a$$

Combine 6a and -7a to get -a.

$$-a+10a^{2}+7+8a^{4}$$

Differentiate w.r.t. a

$$32a^{3}+20a-1$$

Steps Using Derivative Rule for Sum

Add 3 and 4 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}a}(6a+3a^{2}+7+3a^{4}+7a^{2}+5a^{4}-7a)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(6a+10a^{2}+7+3a^{4}+5a^{4}-7a)$$

Combine 3a^{4} and 5a^{4} to get 8a^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(6a+10a^{2}+7+8a^{4}-7a)$$

Combine 6a and -7a to get -a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-a+10a^{2}+7+8a^{4})$$

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}.

$$-a^{1-1}+2\times 10a^{2-1}+4\times 8a^{4-1}$$

Subtract 1 from 1.

$$-a^{0}+2\times 10a^{2-1}+4\times 8a^{4-1}$$

Multiply 2 times 10.

$$-a^{0}+20a^{2-1}+4\times 8a^{4-1}$$

Subtract 1 from 2.

$$-a^{0}+20a^{1}+4\times 8a^{4-1}$$

Multiply 2 times 10.

$$-a^{0}+20a^{1}+32a^{4-1}$$

Subtract 1 from 4.

$$-a^{0}+20a^{1}+32a^{3}$$

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

$$-a^{0}+20a+32a^{3}$$

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

$$-1+20a+32a^{3}$$