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

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

Evaluate

$$6p^{4}+7p^{3}-2p^{2}-9$$

Solution Steps

Combine p^{2} and -3p^{2} to get -2p^{2}.

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

Subtract 5 from -4 to get -9.

$$-2p^{2}+7p^{3}-9+6p^{4}$$

Differentiate w.r.t. p

$$p\left(24p^{2}+21p-4\right)$$

Steps Using Derivative Rule for Sum

Combine p^{2} and -3p^{2} to get -2p^{2}.

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

Subtract 5 from -4 to get -9.

$$\frac{\mathrm{d}}{\mathrm{d}p}(-2p^{2}+7p^{3}-9+6p^{4})$$

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}.

$$2\left(-2\right)p^{2-1}+3\times 7p^{3-1}+4\times 6p^{4-1}$$

Multiply 2 times -2.

$$-4p^{2-1}+3\times 7p^{3-1}+4\times 6p^{4-1}$$

Subtract 1 from 2.

$$-4p^{1}+3\times 7p^{3-1}+4\times 6p^{4-1}$$

Multiply 3 times 7.

$$-4p^{1}+21p^{3-1}+4\times 6p^{4-1}$$

Subtract 1 from 3.

$$-4p^{1}+21p^{2}+4\times 6p^{4-1}$$

Multiply 3 times 7.

$$-4p^{1}+21p^{2}+24p^{4-1}$$

Subtract 1 from 4.

$$-4p^{1}+21p^{2}+24p^{3}$$

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

$$-4p+21p^{2}+24p^{3}$$