Solve (p^3+p^2-7)+(4-4p^3+7p^2)=

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

Evaluate

$$-3p^{3}+8p^{2}-3$$

Solution Steps

Add -7 and 4 to get -3.

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

Combine p^{3} and -4p^{3} to get -3p^{3}.

$$-3p^{3}+p^{2}-3+7p^{2}$$

Combine p^{2} and 7p^{2} to get 8p^{2}.

$$-3p^{3}+8p^{2}-3$$

Differentiate w.r.t. p

$$p\left(16-9p\right)$$

Steps Using Derivative Rule for Sum

Add -7 and 4 to get -3.

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

Combine p^{3} and -4p^{3} to get -3p^{3}.

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

Combine p^{2} and 7p^{2} to get 8p^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(-3p^{3}+8p^{2}-3)$$

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}.

$$3\left(-3\right)p^{3-1}+2\times 8p^{2-1}$$

Multiply 3 times -3.

$$-9p^{3-1}+2\times 8p^{2-1}$$

Subtract 1 from 3.

$$-9p^{2}+2\times 8p^{2-1}$$

Multiply 2 times 8.

$$-9p^{2}+16p^{2-1}$$

Subtract 1 from 2.

$$-9p^{2}+16p^{1}$$

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

$$-9p^{2}+16p$$