# What is the formula of tan2x

## Malle understand math 7, textbook

92 4 Investigating Further Functions Deriving the Tangent Function Theorem Deriving the Tangent Function: f (x) = tanx w f '(x) = 1 _ cos 2 x = 1 + tan 2 x Proof: f (x) = tan x = sinx _ cosx According to the quotient rule: f '(x) = cosx · cosx - sinx · (–sinx) ____ cos 2 x = cos 2 x + sin 2 x __ cos 2 x We can use this term further transform in two ways: 1st type: f '(x) = cos 2 x + sin 2 x __ cos 2 x = 1 _ cos 2 x 2nd type: f' (x) = cos 2 x + sin 2 x __ cos 2 x = 1 + sin 2 x _ cos 2 x = 1 + tan 2 x  Exercise in-depth 4.50 Find the derivative! a) f (x) = x + tanx c) f (x) = x tanx e) f (x) = x (1 + tan x) g) f (x) = tan (2x) b) f ( x) = tanx _ x d) f (x) = x _ tanx f) f (x) = x 2 _ tanx h) f (x) = tanx _ x 2 Derivation of the square root function theorem Derivation of the square root function: f (x) = 9 _ x w f '(x) = 1 _ 2 9 _ x Proof: f (z) - f (x) __ z - x = 9 _ z - 9 _ x _ z - x = 9 _ z - 9 _ x __ (9 _ z) 2 - (9 _ x) 2 = 9 _ z - 9 _ x ___ (9 _ z - 9 x) (9 z + 9 _ x) = 1 _ 9 _ z + 9 _ x f '(x) = lim z ¥ x f (z) - f (x) __ z - x = lim z ¥ x 1 _ 9 _ z + 9 _ x = 1 _ 2 9 _ x  Exercise specialization 4.51 Find f '(x)! a) f (x) = 10 9 _ x b) f (x) = - 9 _ x c) f (x) = x + 9 _ x d) f (x) = 2x - 3 9 _ x 4.52 Find f '(x)! a) f (x) = 1 _ 9 _ x b) f (x) = 1 _ x - 9 _ x c) f (x) = 2 _ 9 _ x + 3 d) f (x) = 9 _ x - 1 _ 9 _ x 4.53 Find the largest possible domain of the function f with f (x) = 9 _ x and examine f with regard to zeros , Monotony, curvature, extreme points and turning points! 4.54 Let a moving body be braked and move for t º 0 according to the time-place function s: t ¦ 9 _ t (t in seconds, s (t) in meters). 1) Set up a formula for the speed of the body at time t! For which t does this formula apply? 2) Set up a formula for the acceleration of the body at time t! For which t does this formula apply? 3) At what point in time is the speed of the body only 1 cm / s? 4) At what point in time is the body's acceleration only -1cm / s 2? For testing purposes only - property of the publisher öbv