# The n th derivative of the function.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.1, Problem 60E

(a)

To determine

## To find: The nth derivative of the function.

Expert Solution

The nth derivative of the function f(x)=xn is n(n1)(n2)...21=n!_.

### Explanation of Solution

Given:

The function is f(x)=xn

Derivative rules:

(1) Constant multiple rule: ddx(cf)=cddx(f)

(2) Power rule: ddx(xn)=nxn1

Calculation:

The first derivative of f(x)=xn is

f(x)=ddx(xn)

By the power rule (2), f(x)=nxn1.

Thus, the first derivative of f(x)=xn is f(x)=nxn1_.

The second derivative of f(x)=xn is,

f(x)=ddx(f(x))=ddx(nxn1)

Apply the constant multiple rule (1) and the power rule (2),

f(x)=nddx(xn1)=n((n1)x(n1)1)=n(n1)xn11=n(n1)xn2

Thus, the second derivative of f(x)=xn is n(n1)xn2_.

The third derivative of f(x)=xn is,

f(x)=ddx(f(x))=ddx(n(n1)xn2)

Apply the constant multiple rule (1) and the power rule (2),

f(x)=n(n1)ddx(xn2)=n(n1)((n2)x(n2)1)=n(n1)(n2)xn21=n(n1)(n2)xn3

Thus, the third derivative of f(x)=xn is n(n1)(n2)xn3_.

From the first, second and the third derivatives of f(x)=xn, the general kth form of f(x)=xn can be expressed as follows,

fk(x)=n(n1)(n2)...(nk+1)xnk

Substitute n for k in the above equation, then the nth derivative of f(x)=xn is

fn(x)=n(n1)(n2)...21xnn=n(n1)(n2)...21x0=n(n1)(n2)...21=n!

Therefore, it can be concluded that the nth derivative of f(x)=xn is n(n1)(n2)...21=n!_.

(b)

To determine

### To find: The nth derivative of the function.

Expert Solution

The nth derivative of f(x)=1x is. (1)nn!xn+1_.

### Explanation of Solution

Given:

The function is f(x)=1x

Derivative rules:

(1) Constant multiple rule: ddx(cf)=cddx(f)

(2) Power rule: ddx(xn)=nxn1

Calculation:

The first derivative of f(x)=1x is obtained as follows.

f(x)=ddx(1x)=ddx(x1)

Apply the power rule (2),

f(x)=(1)x11=x2

Thus, the first derivative of f(x)=1x is f(x)=x2.

The second derivative of f(x)=1x is obtained as follows.

f(x)=ddx(f(x))=ddx(x2)

Apply the constant multiple rule (1) and the power rule (2),

f(x)=(1)ddx(x2)=1((2)x21)=2x3

Thus, the second derivative of f(x)=1x is f(x)=2x3.

The third derivative of f(x)=1x is obtained as follows.

f(x)=ddx(f(x))=ddx(2x3)

Apply the constant multiple rule (1) and the power rule (2),

f(x)=2ddx(x3)=2((3)x31)=6x4

Thus, the third derivative of f(x)=1x is f(x)=6x4.

The first, second and third derivatives of f(x)=1x can be expressed as follows,

f(x)=(1)1!x2f(x)=(1)22!x3f(x)=(1)33!x4

Proceed in the similar way, the nth derivative of f(x)=1x is f(n)(x)=(1)nn!xn+1.

Therefore, it can be concluded that the nth derivative of f(x)=1x is f(n)(x)=(1)nn!xn+1.

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