Solve 2p^3-2p^2+1text{by}2p+1

$$2 p ^ { 3 } - 2 p ^ { 2 } + 1 \text { by } 2 p + 1$$

Differentiate w.r.t. p

$$2\left(by+3p^{2}-2p\right)$$

Steps Using Derivative Rule for Sum

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}.

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

Multiply 3 times 2.

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

Subtract 1 from 3.

$$6p^{2}+2\left(-2\right)p^{2-1}+2byp^{1-1}$$

Multiply 2 times -2.

$$6p^{2}-4p^{2-1}+2byp^{1-1}$$

Subtract 1 from 2.

$$6p^{2}-4p^{1}+2byp^{1-1}$$

Subtract 1 from 1.

$$6p^{2}-4p^{1}+2byp^{0}$$

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

$$6p^{2}-4p+2byp^{0}$$

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

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

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

$$6p^{2}-4p+2by$$

Evaluate

$$2bpy+2p^{3}-2p^{2}+1$$