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:

$$f'(x) =3x^2-8x+5$$

$$f''(x) =6x-8$$

$$f'''(x) =6$$

$$\forall n \in \mathbb{N}, n \geq 4, f^{(n)}(x) =0.$$

Example 6.10.3  

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

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

   
   

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:

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

   
   
   (Check this by induction!)

Example 6.10.4   Let $f$ be the function defined by