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\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

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