Introduction to Mathematical Analysis I - Second Edition

105 Example 4.1.3 Consider the function h : R → R given by h ( x ) = ( 3 x 4 + x + 7 ) 15 . Since h ( x ) is a polynomial we could in principle compute h 0 ( x ) by expanding the power and using Example 4.1.2 . However, Theorem 4.1.5 provides a shorter way. Define f , g : R → R by f ( x ) = 3 x 4 + x + 7 and g ( x ) = x 15 . Then h = g ◦ f . Given a ∈ R , it follows from Theorem 4.1.5 that ( g ◦ f ) 0 ( a ) = g 0 ( f ( a )) f 0 ( a ) = 15 ( 3 a 4 + a + 7 ) 14 ( 12 a 3 + 1 ) . Example 4.1.4 By iterating the Chain Rule, we can extended the result to the composition of more than two functions in a straightforward way. For example, given functions f : G 1 → R , g : G 2 → R , and h : G 3 → R such that f ( G 1 ) ⊂ G 2 , g ( G 2 ) ⊂ G 3 , f is differentiable at a , g is differentiable at f ( a ) , and h is differentiable at g ( f ( a )) , we obtain that h ◦ g ◦ f is differentiable at a and ( h ◦ g ◦ f ) 0 ( a ) = h 0 ( g ( f ( a ))) g 0 ( f ( a )) f 0 ( a ) . Definition 4.1.2 Let G be an open set and let f : G → R be a differentiable function. If the function f 0 : G → R is also differentiable, we say that f is twice differentiable (on G ). The second derivative of f is denoted by f 00 or f ( 2 ) . Thus, f 00 = ( f 0 ) 0 . Similarly, we say that f is three times differentiable if f ( 2 ) is differentiable, and ( f ( 2 ) ) 0 is called the third derivative of f and is denoted by f 000 or f ( 3 ) . We can define in this way n times differentiability and the nth derivative of f for any positive integer n . As a convention, f ( 0 ) = f . Definition 4.1.3 Let I be an open interval in R and let f : I → R . The function f is said to be continuously differentiable if f is differentiable on I and f 0 is continuous on I . We denote by C 1 ( I ) the set of all continuously differentiable functions on I . If f is n times differentiable on I and the nth derivative is continuous, then f is called n times continuously differentiable . We denote by C n ( I ) the set of all n times continuously differentiable functions on I . Exercises 4.1.1 Prove parts (a) and (b) of Theorem 4.1.3 . 4.1.2 Compute the following derivatives directly from the definition. That is, do not use Theo- rem 4.1.3 , but rather compute the appropriate limit directly (see Example 4.1.1 ). (a) f ( x ) = mx + b where m , b ∈ R . (b) f ( x ) = 1 x (here assume x 6 = 0). (c) f ( x ) = √ x (here assume x > 0) 4.1.3 Let f : R → R be given by f ( x ) = ( x 2 , if x > 0; 0 , if x ≤ 0 . (a) Prove that f is differentiable at 0. Find f 0 ( x ) for all x ∈ R . (b) Is f 0 continuous? Is f 0 differentiable? 4.1.4 Let f ( x ) = ( x α , if x > 0; 0 , if x ≤ 0 .