Solve 5p^3-p+2+p^3-4p^2-3p-1

$$5 p ^ { 3 } - p + 2 + p ^ { 3 } - 4 p ^ { 2 } - 3 p - 1$$

Evaluate

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

Solution Steps

Combine 5p^{3} and p^{3} to get 6p^{3}.

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

Combine -p and -3p to get -4p.

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

Subtract 1 from 2 to get 1.

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

Differentiate w.r.t. p

$$18p^{2}-8p-4$$

Steps Using Derivative Rule for Sum

Combine 5p^{3} and p^{3} to get 6p^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(6p^{3}-p+2-4p^{2}-3p-1)$$

Combine -p and -3p to get -4p.

$$\frac{\mathrm{d}}{\mathrm{d}p}(6p^{3}-4p+2-4p^{2}-1)$$

Subtract 1 from 2 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}p}(6p^{3}-4p+1-4p^{2})$$

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}.

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

Multiply 3 times 6.

$$18p^{3-1}-4p^{1-1}+2\left(-4\right)p^{2-1}$$

Subtract 1 from 3.

$$18p^{2}-4p^{1-1}+2\left(-4\right)p^{2-1}$$

Subtract 1 from 1.

$$18p^{2}-4p^{0}+2\left(-4\right)p^{2-1}$$

Multiply 1 times -4.

$$18p^{2}-4p^{0}-8p^{2-1}$$

Subtract 1 from 2.

$$18p^{2}-4p^{0}-8p^{1}$$

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

$$18p^{2}-4p^{0}-8p$$

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

$$18p^{2}-4-8p$$