Higher degree derivatives.

Definition 6.10.1   Let $ f$ be a function, differentiable on an neighborhood of $ x_0$ . If $ f'$ is differentiable at $ x_0$ , its first derivative $ (f')'(x_0)$ is called the the second derivative of $ f$ at $ x_0$ . By induction we define the $ n^{th}$ derivative of a function at a point $ x_0$ ; we denote it by $ f^{(n)}(x_0)$ .

Example 6.10.2   Take $ f(x)=x^3-4x^2+5x-7$ . Then, for every $ x \in \mathbb{R}$ , we have:

$\displaystyle f'(x)$ $\displaystyle =3x^2-8x+5$    
$\displaystyle f''(x)$ $\displaystyle =6x-8$    
$\displaystyle f'''(x)$ $\displaystyle =6$    
$\displaystyle \forall n \in \mathbb{N}, n \geq 4, f^{(n)}(x)$ $\displaystyle =0.$    

Example 6.10.3  
  1. If $ f(x)=\sin x$ , then for every $ n \in \mathbb{N}$ , we have:

    $\displaystyle f^{(n)}(x)= \sin \left( x+ n \frac {\pi}{2} \right).$    

  2. If $ f(x)= e^x$ , then for every $ n \in \mathbb{N}, \; f^{(n)}(x)= e^x$ .
  3. If $ f(x)=\ln x$ , then we have:

    $\displaystyle \forall n \in \mathbb{N}, \; \forall x \in (0,+ \infty ), \; f^{(n)}(x)=\frac {(-1)^{n+1}n!}{x^n}.$    

    (Check this by induction!)

Example 6.10.4   Let $ f$ be the function defined by

   



Subsections
Noah Dana-Picard 2007-12-28