Limites Fundamentais
lim
𝑥→0𝑠𝑒𝑛 𝑥
𝑥=1
lim
𝑥→01−cos𝑥
𝑥=0
lim
𝑥→2𝑥3−23
𝑥−2 =12
lim
𝑥→0𝑡𝑔 𝑥
𝑥=1
lim
𝑥→1𝑥2−1
1−𝑥 = −2
lim
𝑥→0𝑠𝑒𝑛 2𝑥
𝑥=2
lim
𝑥→∞(1+1
𝑥)𝑥=𝑒
lim
𝑥→0 ln𝑥
𝑥−1 =1
lim
𝑥→2𝑥3+23
𝑥+2 = 4
lim
𝑥→0𝑎𝑥−1
𝑥=ln𝑎
lim
𝑥→0(1−cos𝑥)
𝑥2=1
2
lim
𝑥→0𝑠𝑒𝑛 𝑥
2𝑥 =1
2
lim
𝑥→∞ √𝑥
𝑥=1
lim
𝑥→1√𝑥−1
𝑥−1 =1
2
lim
𝑥→3𝑥−3
𝑥=0
lim
𝑥→1𝑥−1
ln𝑥= 1
lim
𝑥→0(1−cos𝑥)
𝑠𝑒𝑛 𝑥 =0
lim
𝑥→0𝑠𝑒𝑛 2𝑥
2𝑥 =1
lim
𝑥→0𝑥2−9
𝑥−3 = 6
lim
𝑥→0𝑡𝑔(𝑥+𝜋)
𝑥=1
lim
𝑥→0𝑒2𝑥 −1
𝑥=2
lim
𝑥→∞𝑥𝑥=∞
lim
𝑥→∞(1+1
𝑥)=𝑒
lim
𝑥→2𝑥2−4
𝑥−2 = 4
lim
𝑥→0(1+𝑥)1
𝑥 = 𝑒
lim
𝑥→∞(1+𝑦
𝑥)𝑥=𝑒𝑦
lim
𝑥→0ln(1+𝑥)1
3𝑥 = 1
3
lim
𝑥→0𝑡𝑔 𝑥−𝑠𝑒𝑛𝑥
𝑥3=1
2
Derivadas de Funções
𝑑𝑓
𝑑𝑥 = lim
∆𝑥→0𝑓(𝑥+∆𝑥)− 𝑓(𝑥)
∆𝑥
𝑑
𝑑𝑥[𝑓(𝑥)]𝑛=𝑛[𝑓(𝑥)]𝑛−1.𝑓′(𝑥)
𝑑𝑓
𝑑𝑥 =𝑑𝑓
𝑑𝑡.𝑑𝑡
𝑑𝑥
𝑑𝑛
𝑑𝑥𝑛[𝑥𝑚]=𝑚!𝑥𝑚−𝑛
(𝑚−𝑛)!
𝑑
𝑑𝑥[𝑓(𝑥)
𝑔(𝑥)]= 𝑓′(𝑥).𝑔(𝑥)−𝑓(𝑥).𝑔′(𝑥)
[𝑔(𝑥)]2
𝑑
𝑑𝑥[𝑓(𝑥).𝑔(𝑥)] =𝑓′(𝑥).𝑔(𝑥)+𝑓(𝑥).𝑔′(𝑥)
𝑑𝑓
𝑑𝑦 =𝑑𝑓
𝑑𝑥.𝑑𝑥
𝑑𝑦
𝑑1
2
𝑑𝑥1
2[𝑥]=√𝑥
1
2!
𝑑𝑓=(𝛿𝑓
𝛿𝑥)𝑑𝑥+(𝛿𝑓
𝛿𝑦)𝑑𝑦
𝑑𝑓=(𝛿𝑓
𝛿𝑦)𝑑𝑦+(𝛿𝑓
𝛿𝑥)𝑑𝑥
𝑑𝑛
𝑑𝑥𝑛[𝑥−𝑚]=(−1)𝑛.Γ(𝑚+ 𝑛).𝑥−(𝑚+𝑛)
Γ(𝑚)
𝑑
𝑑𝑥[𝑐]=0
𝑑
𝑑𝑥[√𝑥]= 1
2√𝑥
𝑑
𝑑𝑥[ln𝑎𝑥]=1
𝑥
𝑑
𝑑𝑥[sen2𝑥]=2cos2𝑥
𝑑
𝑑𝑥[sen𝑥4]=4𝑥3cos𝑥4
𝑑
𝑑𝑥[sen𝑥]2= 2 𝑠𝑒𝑛𝑥 cos𝑥
𝑑
𝑑𝑥[1]=0
𝑑
𝑑𝑥[𝑒𝑛𝑥]= 𝑛𝑒𝑛𝑥
𝑑
𝑑𝑥[ln𝑥]=1
𝑥
𝑑
𝑑𝑥[sen3𝑥]=3cos3𝑥
𝑑
𝑑𝑥[cos𝑥2]=−2𝑥sen𝑥2
𝑑
𝑑𝑥[sen𝑥]3= 3 [𝑠𝑒𝑛𝑥]2cos𝑥
𝑑
𝑑𝑥[2]=0
𝑑
𝑑𝑥[𝑒𝑥]=𝑒𝑥
𝑑
𝑑𝑥[ln2𝑥]=2
2𝑥
𝑑
𝑑𝑥[cos𝑎𝑥]=−𝑎sen𝑎𝑥
𝑑
𝑑𝑥[cos𝑥3]=−3𝑥2sen 𝑥3
𝑑
𝑑𝑥[sen𝑥]4= 4 [𝑠𝑒𝑛𝑥]3cos𝑥
𝑑
𝑑𝑥[𝑥𝑛]=𝑛𝑥𝑛−1
𝑑
𝑑𝑥[𝑒2𝑥]= 2𝑒2𝑥
𝑑
𝑑𝑥[ln3𝑥]=3
3𝑥
𝑑
𝑑𝑥[cos𝑥]=−sen𝑥
𝑑
𝑑𝑥[cos𝑥4]=−4sen𝑥4
𝑑
𝑑𝑥[cos𝑥]2=−2cos𝑥 sen𝑥
𝑑
𝑑𝑥[𝑥1]=1
𝑑
𝑑𝑥[𝑒3𝑥]= 3𝑒3𝑥
𝑑
𝑑𝑥[log𝑏𝑥]=1
𝑥log𝑏𝑒
𝑑
𝑑𝑥[cos2𝑥]=−2sen2𝑥
𝑑
𝑑𝑥[tg𝑥]=1+tg2𝑥
𝑑
𝑑𝑥[cos𝑥]3= −3[cos𝑥]2sen𝑥
𝑑
𝑑𝑥[𝑥2]=2𝑥
𝑑
𝑑𝑥[𝑏𝑥]=𝑏𝑥ln𝑏
𝑑
𝑑𝑥[log2𝑥]=1
𝑥log2𝑒
𝑑
𝑑𝑥[cos3𝑥]=−3sen3𝑥
𝑑
𝑑𝑥[cotg𝑥]=−1−cotg2𝑥
𝑑
𝑑𝑥[cos𝑥]4= −4[cos𝑥]3sen𝑥
𝑑
𝑑𝑥[𝑥3]=3𝑥2
𝑑
𝑑𝑥[2𝑥]=2𝑥ln2
𝑑
𝑑𝑥[𝑠𝑒𝑛 𝑎𝑥]=𝑎cos𝑎𝑥
𝑑
𝑑𝑥[sen𝑥2]=2𝑥cos𝑥2
𝑑
𝑑𝑥[sec𝑥]=tg 𝑥
cos𝑥
𝑑
𝑑𝑥[cosh𝑥]=senh𝑥
𝑑
𝑑𝑥[1
𝑥] =− 1
𝑥2
𝑑
𝑑𝑥[3𝑥]=3𝑥ln3
𝑑
𝑑𝑥[𝑠𝑒𝑛 𝑥]= cos𝑥
𝑑
𝑑𝑥[sen𝑥3]=3𝑥2cos𝑥3
𝑑
𝑑𝑥[cosec𝑥]=−cotg𝑥
sen𝑥
𝑑
𝑑𝑥[senh𝑥]= cosh𝑥
Integrais Básicas
∫xndx=xn+1
n+1
∫𝑑𝑥
2𝑥 =1
2ln2𝑥
∫cos2𝑥 dx = 1
2sen2𝑥
∫𝑑𝑥
𝑎𝑥+𝑏 = 1
𝑎𝑙𝑛|𝑎𝑥+𝑏|
∫𝑥2𝑑𝑥
𝑥2−𝑎2 = x + 𝑎
2𝑙𝑛|𝑥−𝑎
𝑥+𝑎|
∫𝑥dx=x2
2
∫𝑠𝑒𝑛 𝑎𝑥dx= − 1
acos𝑎𝑥
∫sen2𝑎𝑥 dx=x
2− sen2𝑎𝑥
4a
∫𝑑𝑥
𝑥(𝑎𝑥+𝑏) = 1
𝑏𝑙𝑛|𝑥
𝑎𝑥+𝑏|
∫𝑥 𝑑𝑥
𝑥2+𝑎2 = 1
2𝑙𝑛|𝑥2+𝑎2|
∫x2dx=x3
3
∫𝑠𝑒𝑛 𝑥 dx= − cos𝑥
∫𝑐𝑜𝑠2𝑎𝑥 dx =x
2+ sen2𝑎𝑥
4a
∫𝑑𝑥
(𝑎𝑥+𝑏)2 = −1
𝑎(𝑎𝑥+𝑏)
∫𝑥 𝑑𝑥
𝑥2−𝑎2 = 1
2𝑙𝑛|𝑥2−𝑎2|
∫𝑎𝑥dx=ax2
2
∫𝑠𝑒𝑛 2𝑥 dx= − 1
2cos2𝑥
∫ln𝑎𝑥 dx= xln𝑎𝑥−𝑥
∫𝑥3𝑑𝑥
𝑥2+𝑎2 = x2
2− 𝑎2
2𝑙𝑛|𝑥2+𝑎2|
∫𝑑𝑥
𝑥2+𝑎2 = 1
𝑎𝑎𝑟𝑐 𝑡𝑔𝑥
𝑎
∫1
𝑥dx=ln𝑥
∫cos𝑎𝑥 dx= 1
asen𝑎𝑥
∫𝑒𝑏𝑥 dx =ebx
𝑏
∫𝑥3𝑑𝑥
𝑥2−𝑎2 = x2
2+ 𝑎2
2𝑙𝑛|𝑥2−𝑎2|
∫𝑑𝑥
𝑥2−𝑎2 = 1
2𝑎𝑙𝑛|𝑥−𝑎
𝑥+𝑎|
∫𝑑𝑥
𝑎𝑥dx=1
aln𝑎𝑥
∫cos𝑥dx = sen 𝑥
∫𝑥𝑑𝑥
𝑎𝑥+𝑏 = x
𝑎− 𝑏
𝑎2𝑙𝑛|𝑎𝑥+𝑏|
∫𝑥2𝑑𝑥
𝑥2+𝑎2 = x − a arc tgx
𝑎
∫𝑑𝑥
𝑥(𝑥2−𝑎2) = 1
2𝑎2𝑙𝑛|𝑥2+𝑎2
𝑥2|
Identidades Úteis
𝑥2−𝑦2=(𝑥−𝑦)(𝑥+ 𝑦)
log𝑐𝐴.𝐵= log𝑐𝐴+log𝑐𝐵
cosh𝑥 =𝑒𝑥+𝑒−𝑥
2
cos(𝑎+𝑏)=cos𝑎.cos𝑏 − 𝑠𝑒𝑛 𝑎.𝑠𝑒𝑛 𝑏
𝑥3−𝑦3=(𝑥−𝑦)(𝑥2+𝑥𝑦+𝑦2)
log𝑐𝐴
𝐵=log𝑐𝐴−log𝑐𝐵
senh𝑥 =𝑒𝑥−𝑒−𝑥
2
cos(𝑎−𝑏)=cos𝑎.cos𝑏 + 𝑠𝑒𝑛 𝑎.𝑠𝑒𝑛 𝑏
𝑥3+𝑦3=(𝑥+𝑦)(𝑥2−𝑥𝑦+𝑦2)
𝑐𝑜𝑠2ℎ𝑥−𝑠𝑒𝑛2ℎ𝑥 = 1
sen(𝑎+𝑏)=sen𝑎.cos𝑏+cos𝑎 .𝑠𝑒𝑛 𝑏
(𝑥+𝑦)2= 𝑥 2+2𝑥𝑦+𝑦2
log𝑐𝐴= log𝑏𝐴
log𝑏𝐶
𝑠𝑒𝑛2𝑥+𝑐𝑜𝑠2𝑥 =1
sen(𝑎−𝑏)=sen𝑎.cos𝑏−cos𝑎 .𝑠𝑒𝑛 𝑏
(𝑥−𝑦)2= 𝑥 2−2𝑥𝑦+𝑦2
𝑠𝑒𝑐2𝑥= 1+𝑡𝑔2𝑥
cos𝑎 +cos𝑏 = 2cos1
2(𝑎+𝑏) . cos 1
2(𝑎−𝑏)
(𝑥+𝑦)3= 𝑥3+ 3𝑥2𝑦+3𝑥𝑦2+𝑦3
log𝑐𝐴𝐵=𝐵log𝑐𝐴
𝑐𝑜𝑠𝑒𝑐2𝑥 = 1+𝑐𝑜𝑡𝑔2𝑥