Solve (2p^3+6-5p^2)+(7p^3+8+8p^2)

$$( 2 p ^ { 3 } + 6 - 5 p ^ { 2 } ) + ( 7 p ^ { 3 } + 8 + 8 p ^ { 2 } )$$

Evaluate

$$9p^{3}+3p^{2}+14$$

Solution Steps

Combine 2p^{3} and 7p^{3} to get 9p^{3}.

$$9p^{3}+6-5p^{2}+8+8p^{2}$$

Add 6 and 8 to get 14.

$$9p^{3}+14-5p^{2}+8p^{2}$$

Combine -5p^{2} and 8p^{2} to get 3p^{2}.

$$9p^{3}+14+3p^{2}$$

Differentiate w.r.t. p

$$3p\left(9p+2\right)$$

Steps Using Derivative Rule for Sum

Combine 2p^{3} and 7p^{3} to get 9p^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(9p^{3}+6-5p^{2}+8+8p^{2})$$

Add 6 and 8 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}p}(9p^{3}+14-5p^{2}+8p^{2})$$

Combine -5p^{2} and 8p^{2} to get 3p^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(9p^{3}+14+3p^{2})$$

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}.

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

Multiply 3 times 9.

$$27p^{3-1}+2\times 3p^{2-1}$$

Subtract 1 from 3.

$$27p^{2}+2\times 3p^{2-1}$$

Multiply 2 times 3.

$$27p^{2}+6p^{2-1}$$

Subtract 1 from 2.

$$27p^{2}+6p^{1}$$

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

$$27p^{2}+6p$$