# 5. Derivatives

### 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, .