Derivatives of order two or more are called higher derivatives and are represented by the following notation:

The last one is read as “the n^{th} derivative of f with respect to x.”

The definition is given as follows by induction:

**Exercise 1. **Compute the second derivative for the following function:

* Solution*. First we compute the first derivative and then the second derivative for the assigned function:

**Exercise 2. **Compute the second derivative for the following function:

* Solution*. We compute the first derivative and then the second derivative for the assigned function:

**Exercise 3**. Compute the second derivative for the following function:

* Solution*. We compute the first derivative and then the second derivative for the assigned function:

**Exercise 4**. Compute the second derivative for the following function:

* Solution*. We compute before the first derivative and then the second derivative for the assigned function:

** Definition**. A function f with domain D is said to have an absolute maximum at c if f(x) ≤ f(c) for all x D. The number f(c) is called the *absolute maximum* of f on D. The function f is said to have a *local* *maximum* (or *relative* *maximum*) at c if there is some open interval (a, b) containing c and f(c) is the absolute maximum of f on (a, b).

** Definition**. A function f with domain D is said to have an absolute minimum at c if f(x) ≥ f(c) for all x D. The number f(c) is called the absolute minimum of f on D. The function f is said to have a local minimum (or relative minimum) at c if there is some open interval (a, b) containing c and f(c) is the absolute minimum of f on (a, b).

**Definition**. An absolute maximum or absolute minimum of f is called an *absolute* *extremum* of f. A local maximum or minimum of f is called a local **extremum** of f.

** Theorem 1. (Extreme Value Theorem)**. If a function f is continuous on a closed and bounded interval [a,b], then there exist two points, c_{1} and c_{2}, in [a,b] such that f(c_{1}) is the absolute minimum of f on [a,b] and f(c_{2}) is the absolute maximum of f on [a,b].

**Theorem 2**. If f is defined on an open interval (a,b) containing c, f(c) is a local extremum of f and f’(c) exists, then f’(c) = 0.

**Definition**. If f is differentiable at c and f’(c) = 0, then we call c a* critical point or stationary point* of f.

**Definition**. A function f is said to be increasing on an open interval (a,b) if f(x_{1})< f(x_{2}) for all x_{1} and x_{2} in (a,b) such that x_{1}< x_{2}. The function f is said to be *decreasing* on (a,b) if f(x_{1}) > f(x_{2}) for all x_{1} and x_{2} in (a,b) such that x_{1}< x_{2}. The function f is said to be *non*-*decreasing* on (a,b) if f(x_{1}) ≤ f(x_{2}) for all x_{1} and x_{2} in (a,b) such that x_{1}< x_{2}. The function f is said to be non-increasing on (a,b) if f(x_{1})≥ f(x_{2}) for all x_{1} and x_{2} in (a,b) such that x_{1}< x_{2}.

**Theorem 3. **Suppose that two functions f and g are continuous on a closed and bounded interval [a, b] and are differentiable on the open interval (a,b). Then the following statements are true:

(i) If f’(x) > 0 for each x in (a,b), then f is increasing on (a,b).

(ii) If f’(x) < 0 for each x in (a,b), then f is decreasing on (a, b).

(iii) If f’(x) ≥ 0 for each x in (a b), then f is non-decreasing on (a,b).

(iv) If f’(x) ≤ 0 for each x in (a,b), then f is non-increasing on (a,b).

(v) If f’(x) = 0 for each x in (a, b), then f is constant on (a, b).

**Theorem 4. (First Derivative Test for Extremum)**. Let f be continuous on an open interval (a, b) and a < c < b.

(i) If f’(x) > 0 on (a,c) and f’(x) < 0 on (c,b), then f(c) is a local maximum of f on (a,b).

(ii) If f’(x) < 0 on (a,c) and f’(x) > 0 on (c,b), then f(c) is a local minimum of f on (a,b).

**Exercise 5**. Take the function

and locate the intervals where the graph of f is increasing or decreasing, the local extrema and absolute extrema on [-3, 3].

*Solution*. The function f(x) is defined for all real numbers. Hence, its domain is: x [-3, 3]. Now we compute the first derivative to find stationary points and then apply the first derivative test for extrema:

To apply the first derivative test for extrema, we solve the following inequality:

and evaluate the sign of f’.

The function f is decreasing on and is increasing on

Then, applying the first derivative test for extrema, we say:

Now we evaluate the function f at the end points of the interval [-3, 3]:

**Exercise 6.** Take the function and locate the intervals where the graph of f is increasing or decreasing, the local extrema and absolute extrema on function’s domain.

*Solution*. The function f(x) is defined for all real number. Hence, its domain is: **x** ] – ∞, + ∞ [. Now we compute the first derivative to find stationary points and then apply the first derivative test for extrema:

So f’(x) = 0 when 1 – x^{2} = 0. Thus the function f has two critical points:

Now, to apply the first derivative test for extrema, we solve the following inequality:

and evaluate the sign of f’.

Now, applying the first derivative test for extrema, we say:

Moreover, we study the behavior of the function f at the end points of its domain ] -**∞**, + **∞** [:

Showing that the x-axis is a horizontal asymptote for the graph of f.

Now, putting these observations together, we can say that:

**Exercise 7.** Take the function and locate the intervals where the graph of f is increasing or decreasing, the local extrema and absolute extrema on function’s domain.

*1*. Functions and their graph. One–one and onto functions. Composite and inverse functions.

*2*. Linear inequalities. Systems of linear inequalities

*3*. Polynomial inequalities. Rational inequalities. Absolute-value inequalities

*6*. Differentiation. Derivative rules, the chain rule.

*7*. Application of differentiation: L'Hospital's Rule

*8*. Vertical, Horizontal and Slant asymptotes

*9*. Higher Order Derivatives. Applications of differentiation: local and absolute extremes of a function

*10*. Curve Sketching

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion