Single Variable 미적분 토막 연습

ref:

Wiki: https://en.wikipedia.org/wiki/Lists_of_integrals#Lists_of_integrals

Differentiation Rules : https://en.wikipedia.org/wiki/Differentiation_rules

■ graphs of exponential functions f(x) = a^x , a > 0.  Then f(x) is a continuous function with domain R, and range (0, ∞), and f(x) > 0 for all x.   

When x, y are real numbers, and a, b > 0

■ what is 'e'?

 

* derivative of an exponential function is proportional to the function itself. 

What is a base 'a' of the function f(x) = a^x  so that f'(x) = f(x). It will make differentiation very nice looking.

e is that number.  e is an irrational number about 2.71818

So, of all the possible exponential functions f(x) = a^x that pass (0, 1), the function f(x) = e^x is the one whose derivative value(slope) at x=0 is 1.  Because e^x's derivative is e^x itself, its height is its slope. 

■

ex)

■

ex)

■  Log Functions

 

■  Derivatives of Log Functions

since

example)

--------

Thus f'(x) = 1/x for all x ≠ 0

■  Derivatives and Integrations 

ex) Differentiate y using log differentiations

 

taking ln's to both sides,

Differentiating implicitely w.r.t. x,

 

ex)  y = f(x)^g(x)  형태의 미분은? 

Since 

예) 

또는,

하여

  

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