# 2. Derivatives

Part 1.: problems, solutions.
Part 2.: problems, solutions.
Part 3.: problems, solutions.

### Definition

We say that a function is differentiable in point , if there exists the limit:

and is finite. Then this limit is called the derivative of in and denoted as .

E.g. function is differentiable in , because:

so: .

Function is not differentiable in , because:

so this limit is infinite.

Function is not differentiable in , because:

but

so the limit we need does not exist.

### Derivatives of some simple functions

It is easy to calculate the following facts. They are useful to calculate derivatives of more complicated functions. Let

### Arithmetic of derivatives

The arithmetic theorem of limits of functions implies immediately that:

if respective derivatives exist.

It is also easy to notice that in the case of multiplication it is not so easy — multiplying the expression in the definition does not give the expression for derivative of the product. Nevertheless, it is easy to check that:

and

if respective derivatives exist.

E.g. let . Then: .

### Composition of functions

We have the following theorem. Given functions and , we have , if the derivatives of in and of in exist.

Therefore e.g. .

### Local extrema and intervals of monotonicity

Local minimum or respectively local maximum of a function is an argument , such that there exists an interval , such that is the least or respectively the greatest value of the function in this interval.

We have the following theorem: if a function has in local extreme and the derivative, then . Attention: the reverse implication does not hold, so the points in which the derivative equals zero are merely the candidates for extrema.

E.g. has minimum in and indeed , so . On the other hand for we get so again , but for the function does not have a local extreme.

We say that a function is strictly increasing (respectively non-decreasing, strictly decreasing, non-increasing) on an interval if for any , such that , we have (respectively: , , ).

The following theorem holds: if (respectively: , , ) for any , then is strictly increasing (respectively non-decreasing, strictly decreasing, non-increasing) on .

E.g. for we have for and for . So is strictly decreasing on and strictly increasing on .

Therefore a function continuous in has in local maximum (respectively minimum), if there exists an interval , such that is differentiable in its every point, and (respectively ) for and (respectively ) for .

E.g. is the local minimum of , given what we calculated above about its derivative.

### Derivative of the inverse function

If is strictly monotone and continuous, then if derivative exists in and is non-zero.

E.g. let . Then . Therefore:

### Geometric interpretation — tangent line

It is easy to see that the derivative of a function is the limit of tangents of the angles of secant lines intersecting the graph of the function in and . Therefore, it is the tangent of the angle of the tangent line to the graph in . Therefore, if is continuous in , then is the equation of the tangent line to in the point .

E.g.: . Therefore the tangent line to this parabola in is .

### Rolle and Lagrange theorems

Rolle theorem states that if is continuous on and differentiable on and , then there exists , such that .

Therefore Lagrange theorem holds: if is continuous on and differentiable on , then there exists , such that .

### L’Hôpital’s rule

L’Hôpital’s rule makes it possible to calculate some difficult cases in arithmetic of limits of functions using derivatives. If or and also and are defined on for some , then , if the second limit exists.

E.g. we calculate . We have and . Also and . Therefore, .

### Higher-order derivatives

Obviously, if the derivative of a function is differentiable, we can calculate its derivative, the derivative of the derivative, , which describes how the first derivative changes. And further we can calculate the third, fourth, etc. derivatives. Generally -th derivative will be denoted by and .

E.g., for , we get:

and for any ,

### Condition for the existence of a local extremum

The following theorem holds. If for some :

• is differentiable in up to at least -th derivative,
• ,
• (respectively: ),

then has a minimum (respectively, maximum) in .

E.g. has maximum in , because:

### Convex and concave functions, inflection points

We shall say that is convex on an interval (a,b), if its graph between any two points is under its secant line given by those points. In other words if for any and :

If the reverse inequality holds, the function is concave. If is convex on and concave on (or reversely) and continuous in , we will say that is an inflection point.

The following theorems hold:

• if exists on a given interval and is increasing (respectively, decreasing), then is convex (respectively, concave) on this interval,
• if exists on a given interval and is always positive (respectively, negative), then is convex (respectively, concave) on this interval,

E.g.: . We get , is positive on so is convex on this interval and negative on — so is concave on it. Point is an inflection point.

### Asymptotes

Asymptotes are lines which are lines to which the diagram of a function converges. Asymptotes can be vertical, horizontal or oblique.

If , then line is a right vertical asymptote. Analogously, if , this line is a left vertical asymptote. E.g. oraz , so asymptote is a vertical asymptote of this function.

If or , then line is a respectively right or left horizontal asymptote of this function. Since , is a horizontal asymptote of .

A line is an oblique asymptote (respectively left or right), if or . If such a line is asymptote (assume it is a right asymptote), then oraz .

E.g.: let , then and . Therefore, is an oblique asymptote of this function.