Application of Derivatives

y = f(x), point P(a,f(a))
Tangent equation: y – f(a) = f'(a) (x-a)

Slope of normal: -${1}/{{dy}/{dx}}$

Formula of approximation:
f(a+h) ≑ f(a) + h . f'(a)

Rolle’s Theorem:

Function f is continuous & differentiable on (a,b) and f(a) = f(b)

Then there exists at least one point c∊ (a,b) such that f'(c) = 0

Langrange’s Mean value theorem:
${f(b) – f(a)}/{b – a} = f'(c) $

A function:

Increasing -> f'(x) > 0 , and maximum -> f'(c) = 0 & f”(c) <0
Decreasing -> f'(x) < 0 , and minimum -> f'(c) = 0 & f”(c) > 0

1st Derivative Test:

Maximum: f'(c-h) > 0 and f'(c+h)<0
Minimum: f'(c-h) < 0 and f'(c+h)>0