Rules of derivation

Theme: Functions

Requires: Derivative, Limits of functions, Newton's binomial

- Derivative of the function "integer power", using Newton's binomial.

$$(x+a)^n = x^n + nax^(n-1) + ...$$

therefore, if $$a\to 0$$, $$(x^n)'=nx^(n-1)$$

- Derivative of the sum, substraction and multiplication times a number: "linearity"... by now we only can do polynomials!!

- What about product of functions? Leibniz Rule,

$$f(x+h)g(x+h) \approx (f(x)+hf'(x))(g(x)+hg'(x))=f(x)g(x) + hfg' + hf'g + h^2 f'g'...$$

let's check whether it works with two monomials

$$(x^5)'=(x^2\cdot x^3)'=2x\cdot x^3 + x^2\cdot 3x^2= 5x^4$$

- What happens when the power is not an integer? Case of the square root or 1/x... we accept by now the extension of the rule...

- Exponential function, special value of "e"

- And the logarithm? Let's check the inverse function...

- And the compound function? Chain rule,

$$f(g(x+h))\approx f(g(x)+hg'(x)) \approx f(g(x))+hg'(x)\cdot f'(g(x))$$

What does it really mean?

Let's check a simple example:

$$x=exp(log(x)) -> ...$$

- Now, with exp and log, let's return to the power of real exponent

$$x^a= exp(a\log(x)) -> exp(a\cdot\log(x)) a/x = x^a a/x =ax^(a-1)$$

OK! It's coherent!!

- Go on with trigonometric functions

starting with $$\sin(h)\approx h$$ when $$h\to 0$$,

$$sin(x+h)=sin(x)cos(h) + cos(x)sin(h) -> sin(x)+h cos(x)$$

- Summary of rules and applications

Now read: Integration, Taylor series, 1D Optimizacin

New terms: Leibniz rule, Chain rule