Solve P(6)=8!(2)6!x

$$P ( 6 ) = 8 ! ( 2 ) 6 ! x$$

Evaluate

$$58060800x$$

Solution Steps

The factorial of 8 is 40320.

$$40320\times 2\times 6!x$$

Multiply 40320 and 2 to get 80640.

$$80640\times 6!x$$

The factorial of 6 is 720.

$$80640\times 720x$$

Multiply 80640 and 720 to get 58060800.

$$58060800x$$

Differentiate w.r.t. x

$$58060800$$

Steps Using Definition of a Derivative

The factorial of 8 is 40320.

$$\frac{\mathrm{d}}{\mathrm{d}x}(40320\times 2\times 6!x)$$

Multiply 40320 and 2 to get 80640.

$$\frac{\mathrm{d}}{\mathrm{d}x}(80640\times 6!x)$$

The factorial of 6 is 720.

$$\frac{\mathrm{d}}{\mathrm{d}x}(80640\times 720x)$$

Multiply 80640 and 720 to get 58060800.

$$\frac{\mathrm{d}}{\mathrm{d}x}(58060800x)$$

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

$$58060800x^{1-1}$$

Subtract 1 from 1.

$$58060800x^{0}$$

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

$$58060800\times 1$$

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

$$58060800$$