#### To determine

**To find:**

The value of d9dx9(x8lnx).

#### Answer

8!x

#### Explanation

**Given:**

The expression d9dx9(x8lnx)

**Formula used:**

ddx(f(x)g(x))=ddx(f(x))×g(x)+f(x)ddx(g(x))ddx(xn)=nxn−1

Use product rule of differentiation ddx(f(x)g(x))=ddx(f(x))×g(x)+f(x)ddx(g(x)) to evaluate the given expression as follows:

ddx[x8lnx]=8x7lnx+x8x−1=8x7lnx+x7d2dx2[x8lnx]=8×7x6lnx+8x7x−1+7x6=8×7x6lnx+(8+7)x6

Differentiate again both sides with respect to *x* and get:

d3dx3[x8lnx]=8×7×6x5lnx+8×7x5+15×5x5=(8×7×6)x5lnx+(56+75)x5

When we differentiate 5 more times (that is the 8^{th} differential) the above expression, the second term of the expression will come out to be a constant. Thus, the second term of the 9^{th} differential of the given expression would be 0.

Also, in the 8^{th} differential of the given expression, the first term of the expression would be (8×7×6×5×4×3×2)lnx. Thus, the first term of the 9^{th} differential of the given expression would be (8×7×6×5×4×3×2)×1x=8!x.

Hence,

d9dx9(x8lnx)=8!x+0=8!x

**Conclusion:**

Hence, the value of d9dx9(x8lnx)=8!x.