
1. Sin x'in Türevi
Sin x'in türevi cos x'tir.
a.
(sin x)′=(x)′ cos x
(sin x)′=1 cos x
(sin x)′=cos x
b.
(sin ax)′=(ax)′ cos ax
(sin ax)′=a cos ax
c.
(sin u)′=u′ cos u
Soru
f(x)=sin (x2+3x−5)⇒f′(x)= ?
Cevap
(sin u)′=u′ cos u
[sin (x2+3x−5)]′=(x2+3x−5)′ cos (x2+3x−5)
[sin (x2+3x−5)]′=(2x+3) cos (x2+3x−5)
2. Cos x'in Türevi
Cos x'in türevi -sin x'tir.
a.
(cos x)′=−(x)′ sin x
(cos x)′=−1 sin x
(cos x)′=−sin x
b.
(cos ax)′=−(ax)′ sin ax
(cos ax)′=−a sin ax
c.
(cos u)′=−u′ sin u
Soru
f(x)=cos (−3x3+5x)⇒f′(x)= ?
Cevap
(cos u)′=−u′ sin u
[cos (−3x3+5x)]′=−(−3x3+5x)′ sin (−3x3+5x)
[cos (−3x3+5x)]′=−(−3.3.x2+5) sin (−3x3+5x)
[cos (−3x3+5x)]′=−(−9x2+5) sin (−3x3+5x)
[cos (−3x3+5x)]′=(9x2−5) sin (−3x3+5x)
3. Tan x'in Türevi
Tan x'in türevi 1+tan² x'dir.
a.
(tan x)′=(x)′ (1+tan2 x)
(tan x)′=1 (1+tan2 x)
(tan x)′=1+tan2 x
b.
(tan ax)′=(ax)′ (1+tan2 ax)
(tan ax)′=a (1+tan2 ax)
c.
(tan u)′=u′ (1+tan2 u)
Soru
f(x)=tan (3x+5)⇒f′(x)= ?
Cevap
(tan u)′=u′ (1+tan2 u)
[tan (3x+5)]′=(3x+5)′ [1+tan2 (3x+5)]
[tan (3x+5)]′=3 [1+tan2 (3x+5)]
[tan (3x+5)]′=3.1+3.tan2 (3x+5)
[tan (3x+5)]′=3+3 tan2 (3x+5)
4. Cot x'in Türevi
Cot x'in türevi -(1+cot² x)'dir.
a.
(cot x)′=−(x)′ (1+cot2 x)
(cot x)′=−1 (1+cot2 x)
(cot x)′=−(1+cot2 x)
b.
(cot ax)′=−(ax)′ (1+cot2 ax)
(cot ax)′=−a (1+cot2 ax)
c.
(cot u)′=−u′ (1+cot2 u)
Soru
f(x)=cot x2⇒f′(x)= ?
Cevap
(cot u)′=−u′ (1+cot2 u)
(cot x2)′=−(x2)′ (1+cot2 x2)
(cot x2)′=−2x (1+cot2 x2)
(cot x2)′=−2x.1−2x.cot2 x2
(cot x2)′=−2x−2x cot2 x2
5. Sec x'in Türevi
a.
(sec x)′=(x)′ sec x tan x
(sec x)′=1 sec x tan x
(sec x)′=sec x tan x
b.
(sec ax)′=(ax)′ sec ax tan ax
(sec ax)′=a sec ax tan ax
c.
(sec u)′=u′ sec u tan u
Soru
f(x)=sec 3x⇒f′(x)= ?
Cevap
(sec u)′=u′ sec u tan u
(sec 3x)′=(3x)′ sec 3x tan 3x
(sec 3x)′=3 sec 3x tan 3x
6. Csc x'in Türevi
a.
(csc x)′=−(x)′ csc x cot x
(csc x)′=−1 csc x cot x
(csc x)′=−csc x cot x
b.
(csc ax)′=−(ax)′ csc ax cot ax
(csc ax)′=−a csc ax cot ax
c.
(csc u)′=−u′ csc u cot u
Soru
f(x)=csc 4x⇒f′(x)= ?
Cevap
(csc u)′=−u′ csc u cot u
(csc 4x)′=−(4x)′ csc 4x cot 4x
(csc 4x)′=−4 csc 4x cot 4x