   Chapter 3.6, Problem 54E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find d 9 d x 9 ( x 8 ln x ) .

To determine

To find: The derivative of d9dx9(x8lnx).

Explanation

Let f(x)=x8lnx, then d9dx9(f(x))=d9dx9(x8lnx).

The first derivative of f(x) is computed as follows,

ddx(f(x))=ddx(x8lnx)

Apply the product rule ddx(fg)=fddx(g)+gddx(f),

ddx(f(x))=x8ddx(lnx)+lnxddx(x8)=x8(1x)+lnx(8x81)=x7+lnx(8x7)=8x7lnx+x7

Thus, the first derivative of f(x) is ddx(f(x))=8x7lnx+x7.

The second derivative of f(x) is computed as follows,

d2dx2(f(x))=ddx(ddx(f(x)))=ddx(8x7lnx+x7)=ddx(8x7lnx)+ddx(x7)=ddx(8x7lnx)+(7x71)

Apply the product rule ddx(fg)=fddx(g)+gddx(f),

d2dx2(f(x))=[8x7ddx(lnx)+lnxddx(8x7)]+(7x6)=[8x7(1x)+lnx(87x71)]+(7x6)=[8x6+lnx(87x6)]+(7x6)=87x6lnx+(8+7)x6

Thus, the second derivative of f(x) is d2dx2(f(x))=87x6lnx+(8+7)x6

The third derivative of f(x) is computed as follows,

d3dx3(f(x))=ddx(87x6lnx+(8+7)x6)=ddx(87x6lnx)+ddx((8+7)

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