[미적분] calculus(미적분) differential calculus(미분)

4. Differential Calculus

Definition of Derivative. The derivative f`(x) is defined by the equation
f`(x) = , provided the limit exists. The number f`(x) is also called the rate of change of f at x.

Hint. an - bn = (a-b)
Hint. sin x - sin y = 2 sin cos
Hint. = 1
Hint. cos x - cos y = -2 sin sin
Example 7. Continuity of functions having derivatives. If a function f has a derivative at a point x, then it is also continuous at x (반대는 성립 X일수도)
- f(x+h) = f(x) + h()
- Continuity : (a) f is defined at p
(b)

Theorem 4.1. Let f and g be two functions defined on a common interval. At each point where f and g have a derivative, the same is true of the sum f+g, the difference f-g, the product f ? g, and the quotient f/g. (For f/g we need the extra proviso that g is not zero at the point in question.) The derivatives of these functions are given by the following formulas :
(i) (f + g)` = f` + g`
(ii) (f - g)` = f` - g`
(iii)…