Solve 25*25*25*25*(-p)*(-p)*(-p)*(-p)*(-p)

$$25 \cdot 25 \cdot 25 \cdot 25 \cdot ( - p ) \cdot ( - p ) \cdot ( - p ) \cdot ( - p ) \cdot ( - p )$$

Evaluate

$$-390625p^{5}$$

Solution Steps

Multiply -p and -p to get \left(-p\right)^{2}.

$$25\times 25\times 25\times 25\left(-p\right)^{2}\left(-p\right)\left(-p\right)\left(-p\right)$$

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

$$25\times 25\times 25\times 25\left(-p\right)^{3}\left(-p\right)\left(-p\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$25\times 25\times 25\times 25\left(-p\right)^{4}\left(-p\right)$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$25\times 25\times 25\times 25\left(-p\right)^{5}$$

Multiply 25 and 25 to get 625.

$$625\times 25\times 25\left(-p\right)^{5}$$

Multiply 625 and 25 to get 15625.

$$15625\times 25\left(-p\right)^{5}$$

Multiply 15625 and 25 to get 390625.

$$390625\left(-p\right)^{5}$$

Expand \left(-p\right)^{5}.

$$390625\left(-1\right)^{5}p^{5}$$

Calculate -1 to the power of 5 and get -1.

$$390625\left(-1\right)p^{5}$$

Multiply 390625 and -1 to get -390625.

$$-390625p^{5}$$

Differentiate w.r.t. p

$$-1953125p^{4}$$

Steps Using Definition of a Derivative

Multiply -p and -p to get \left(-p\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(25\times 25\times 25\times 25\left(-p\right)^{2}\left(-p\right)\left(-p\right)\left(-p\right))$$

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

$$\frac{\mathrm{d}}{\mathrm{d}p}(25\times 25\times 25\times 25\left(-p\right)^{3}\left(-p\right)\left(-p\right))$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}p}(25\times 25\times 25\times 25\left(-p\right)^{4}\left(-p\right))$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}p}(25\times 25\times 25\times 25\left(-p\right)^{5})$$

Multiply 25 and 25 to get 625.

$$\frac{\mathrm{d}}{\mathrm{d}p}(625\times 25\times 25\left(-p\right)^{5})$$

Multiply 625 and 25 to get 15625.

$$\frac{\mathrm{d}}{\mathrm{d}p}(15625\times 25\left(-p\right)^{5})$$

Multiply 15625 and 25 to get 390625.

$$\frac{\mathrm{d}}{\mathrm{d}p}(390625\left(-p\right)^{5})$$

Expand \left(-p\right)^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}p}(390625\left(-1\right)^{5}p^{5})$$

Calculate -1 to the power of 5 and get -1.

$$\frac{\mathrm{d}}{\mathrm{d}p}(390625\left(-1\right)p^{5})$$

Multiply 390625 and -1 to get -390625.

$$\frac{\mathrm{d}}{\mathrm{d}p}(-390625p^{5})$$

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

$$5\left(-390625\right)p^{5-1}$$

Multiply 5 times -390625.

$$-1953125p^{5-1}$$

Subtract 1 from 5.

$$-1953125p^{4}$$