Solve {l}{-(-3m)(m^2)-3(m^3)}{-5(-2p^4)(-p^4)}

$$\left. \begin{array} { l } { - ( - 3 m ) ( m ^ { 2 } ) - 3 ( m ^ { 3 } ) } \\ { - 5 ( - 2 p ^ { 4 } ) ( - p ^ { 4 } ) } \end{array} \right.$$

Evaluate

$$-10p^{8}$$

Solution Steps

The opposite of -3m is 3m.

$$3mm^{2}-3m^{3}-5\left(-2\right)p^{4}\left(-p^{4}\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$3m^{3}-3m^{3}-5\left(-2\right)p^{4}\left(-p^{4}\right)$$

Combine 3m^{3} and -3m^{3} to get 0.

$$0-5\left(-2\right)p^{4}\left(-p^{4}\right)$$

Multiply 5 and -2 to get -10.

$$0-\left(-10p^{4}\left(-p^{4}\right)\right)$$

Multiply -10 and -1 to get 10.

$$0-10p^{4}p^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$0-10p^{8}$$

Anything plus zero gives itself.

$$-10p^{8}$$

Differentiate w.r.t. p

$$-80p^{7}$$

Steps Using Definition of a Derivative

The opposite of -3m is 3m.

$$\frac{\mathrm{d}}{\mathrm{d}p}(3mm^{2}-3m^{3}-5\left(-2\right)p^{4}\left(-p^{4}\right))$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}p}(3m^{3}-3m^{3}-5\left(-2\right)p^{4}\left(-p^{4}\right))$$

Combine 3m^{3} and -3m^{3} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}p}(0-5\left(-2\right)p^{4}\left(-p^{4}\right))$$

Multiply 5 and -2 to get -10.

$$\frac{\mathrm{d}}{\mathrm{d}p}(0-\left(-10p^{4}\left(-p^{4}\right)\right))$$

Multiply -10 and -1 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}p}(0-10p^{4}p^{4})$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}p}(0-10p^{8})$$

Anything plus zero gives itself.

$$\frac{\mathrm{d}}{\mathrm{d}p}(-10p^{8})$$

The derivative of ax^{n} is nax^{n-1}.

$$8\left(-10\right)p^{8-1}$$

Multiply 8 times -10.

$$-80p^{8-1}$$

Subtract 1 from 8.

$$-80p^{7}$$