   Chapter 3.3, Problem 52E ### 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 the given derivative by finding the first few derivatives and observing the pattern that occurs. d 35 d x 35 ( x sin x )

To determine

To find: The derivative of d35dx35(xsinx).

Explanation

Derivative rules used:

(1) Product Rule:ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

(2) Sum Rule :ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)]

(3) Constant Multiple Rule :ddx[cf(x)]=cddx[f(x)]

Calculation:

Let f(x)=xsinx (1)

Substitute h(x)=sinx in equation (1),

Obtain the first derivative of f(x).

f(x)=ddx[xh(x)]

Apply the product rule (1) and simplify further,

f(x)=xddx(h(x))+h(x)ddx(x)=xh(x)+h(x)(1)=xh(x)+h(x)

Thus, the first derivative of f(x) is f(x)=xh(x)+h(x).

Obtain the second derivative of f(x).

f(x)=d2dx2(f(x))=ddx(f(x))=ddx(xh(x)+h(x))

Apply the derivative rules (1), (2) and (3),

f(x)=ddx(xh(x))+ddx(h(x))=xddx(h(x))+h(x)ddx(x)+ddxh(x)=x(h(x))+h(x)(1)+h(x)=xh(x)+2h(x)

Thus, the second derivative of f(x) is f(x)=xh(x)+2h(x).

Obtain the third derivative of f(x).

f(x)=d3dx3(f(x))=ddx(f(x))=ddx(xh(x)+2h(x))

Apply the derivative rules (1) and (2),

f(x)=ddx(xh(x))+ddx(2h(x))=[xddx(h(x))+h(x)ddx(x)]+2ddx[h(x)]=x(h(x))+h(x)(1)+2h(x)=xh(x)+3h(x)

Thus, the third derivative of f(x) is f(x)=xh(x)+3h(x).

Proceed in the similar way, the nth derivative of f(x) is f(n)(x)=xh(n)(x)+nh(n1)(x)

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