Calculus Of One Real Variable  By Pheng Kim Ving Chapter 4: More On The Derivative  Section 4.1: Higher-Order Derivatives 4.1 Higher-Order Derivatives

 1. Mathematical Induction

## Putting To Practice

Example 1.1

Solution

EOS

 2. Higher-Order Derivatives

Let y = f(x) be a differentiable function. If the derivative y' = f '(x) is itself differentiable, its derivative is called the
second derivative of y = f(x) and is denoted y'' or f ''(x) or d 2y/dx2 or (d 2/dx2) f(x):

 3. Factorials

Let n be a positive integer. The factorial of n is defined as the product of all integers from 1 thru to n and is denoted by
n!, read n factorial, so that n! = 1 Χ 2 Χ 3 Χ ... Χ n. It's a product of n factors, hence the name factorial. The factorial of
0 is defined to be 1: 0
! = 1 (there's a reason for this definition; it's not needed here). We have:

The factorial expansion is also written in decreasing order of the factors:

n! = n Χ (n  1) Χ (n  2) Χ  Χ 2 Χ 1.

 4. Derivatives Of All Orders And Mathematical Induction

Example 4.1

Let f(x) = 1/x. Find enough derivatives of different orders of f to enable you to guess the general formula for f (n)(x),
where n is a positive integer. Then use mathematical induction to prove your guess.

Solution
We have:

EOS

Note the use of (1)n to specify the sign. If n is even then we get the  + sign; if n is odd then we get the    sign. For
(1)n+1, if n is even then n + 1 is odd, hence we get the    sign; if n is odd then n + 1 is even, hence we get the 
sign.

 Problems & Solutions

1.  Prove the following formula by using mathematical induction:

where n is any positive integer. Evaluate:

1 + 2 + ... + 5,000.

## Solution

For n = 1 we have:

2.  Let f and g be twice-differentiable functions. Prove that ( fg)'' = f ''g + 2 f 'g' + fg''.

## Solution

( fg)''   =   (( fg)')'   =   ( f 'g + fg' )'   =   ( f 'g)' + ( fg' )'   =   ( f ''g + f 'g' ) + ( f 'g' + fg'' )   =   f ''g + 2 f 'g' + fg''.

3.  Let f(x) = 1/(x + 2). Calculate enough derivatives of different orders of f to enable you to guess the general formula
for f (n)(x), where n is any positive integer. Then use mathematical induction to prove your guess.

Solution

We have f(x) = (x + 2)1. Then:

Solution

5.  Let f(t) = t2/3. Calculate enough derivatives of different orders of f to enable you to guess the general formula for

f (n)(t), where n is any positive integer. Then use mathematical induction to prove your guess.

Solution

We have: