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Calculus 1  Problems & Solutions  –  Chapter 2  –  Section 2.4

 

2.4
Higher-Order Derivatives

 

 

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Review

 

1. Mathematical Induction

 

Principle Of Mathematical Induction

 

 

Putting To Practice

 

 

Example 1.1

 

 

Solution

EOS

 

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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 d2y/dx2 or (d2/dx2) f(x):

 

 

 

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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 ... Χn. 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.

 

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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.

 

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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:

 

 

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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''.

 

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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:

 

 

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Solution

 

 

 

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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:

 

 

 

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