Next: 11.2 Some General Differentiation Up: 11. Calculation of Derivatives Previous: 11. Calculation of Derivatives   Index

# 11.1 Derivatives of Some Special Functions

11.1   Theorem (Derivative of power functions.) Let and let
. Here

Let be an interior point of domain. Then is differentiable at , and

If and we interpret to be .

Proof: First consider the case . For all in domain we have

Let be a generic sequence in domain such that . Let . Then and hence by theorem 7.10 we have and hence

This proves the theorem in the case . If then (since for other values of , is not an interior point of domain). In this case

Hence

Thus in all cases the formula holds.

11.2   Corollary (Of the proof of theorem 11.1) For all ,

11.3   Theorem (Derivatives of and .) Let and let
for all . Then and are differentiable on , and for all

 (11.4) (11.5)

Proof: If the result is clear, so we assume . For all and all , we have

(Here I've used an identity from theorem 9.21.) Let be a generic sequence in such that . Let and let . Then so by lemma 9.34 we have . Also , and for all , so by (9.38), . Hence

and this proves formula (11.4).

The proof of (11.5) is similar.

11.6   Exercise. A Prove that if , then

11.7   Theorem (Derivative of the logarithm.) The logarithm function is differentiable on , and

Proof: Let , and let . Then

Case 1: If then represents the area of the shaded region in the figure.

We have

so by monotonicity of area

Thus
 (11.8)

Case 2. If we can reverse the roles of and in equation (11.8) to get

or

In both cases it follows that

Let be a generic sequence in such that . Then , so by the squeezing rule

i.e.

Hence

We have proved that .

11.9   Assumption (Localization rule for derivatives.) Let be two real valued functions. Suppose there is some and such that

and such that

If is differentiable at , then is differentiable at and .

This is another assumption that is really a theorem, i.e. it can be proved. Intuitively this assumption is very plausible. It says that if two functions agree on an entire interval centered at , then their graphs have the same tangents at .

11.10   Theorem (Derivative of absolute value.) Let for all . Then for all and is not defined.

Proof: Since

it follows from the localization theorem that

To see that is not differentable at , we want to show that

does not exist. Let . Then , but and we know that does not exist. Hence does not exist, i.e., is not differentiable at .

11.11   Definition ( notation for derivatives.) An alternate notation for representing derivatives is:

or

This notation is used in the following way

Or:

Let . Then .

Let . Then .

The notation is due to Leibnitz, and is older than our concept of function.

Leibnitz wrote the differentiation formulas as  ," or if , then  " The notation for derivatives is due to Joseph Louis Lagrange (1736-1813). Lagrange called the derived function of and it is from this that we get our word derivative. Leibnitz called derivatives, differentials and Newton called them fluxions.

Many of the early users of the calculus thought of the derivative as the quotient of two numbers

when was infinitely small''. Today infinitely small'' real numbers are out of fashion, but some attempts are being made to bring them back. Cf Surreal Numbers : How two ex-students turned on to pure mathematics and found total happiness : a mathematical novelette, by D. E. Knuth.[30]. or The Hyperreal Line by H. Jerome Keisler[28, pp 207-237].

Next: 11.2 Some General Differentiation Up: 11. Calculation of Derivatives Previous: 11. Calculation of Derivatives   Index
Ray Mayer 2007-09-07