DerivativesWorksheet
CalculusI
Findthederivativeofeachfunction.Simplifyallresultstoreasonablynicefunctions,unlessotherwisestated.
(
1. ! " = " %& + " ) − 7" + , - − " + " .) )
2. / =
0
2
+
-
12
1
13% )1.(
3. / =
(1 4
4. ! " = 3 tan " − 5 sec " +
5. D " =
6. ℎ " =
E F 3GH 1
1 I 3%
=>? 1
%@
− 2 arcsin "
1 I ?KH 131 =>?I 1
1 =>? 1
0
(1
7. / = 3" − 1
+M
− ln " - + 1 − arctan 4" + 3 I
8. ! P = sin( M Q − cos P 9. / = arccos
1
13%
1
+ M .1 csc "
10. ! " = log ( ( 5 ) + log ) ( ") − 7?V= 1 WX
Foreachcurve,find .
W1
11. " - + 2"/ − 3/ - = 5
1
12. ln
= −10"
X
Ineachcase,findtheequationofalltangentlinestothegivencurveatthegiven"-value.
13. / = 2" + 1 ) at" = 1
14. " - / − / - " = 2at" = −1
Findthe"-valuesatwhichthecurvehasahorizontaltangentline.
15. / = arctan(M -1 − ")
16. / =
2
) 1
1 I 30
Foreachfunction,sketchapossiblegraphofthederivative.
17.
18.
Derivatives Worksheet Answers
Calculus I
1. 𝑓𝑓 ′ (𝑥𝑥) = 17𝑥𝑥16 + 5𝑥𝑥 2 − 7 − 2
2. 𝑦𝑦 ′ =
−4
3
3 √𝑥𝑥 4
6
6
1
√𝑥𝑥
− 𝑥𝑥 4
6
4
3. 𝑦𝑦 ′ = − 5𝑥𝑥 3 + 5𝑥𝑥 4 + 𝑥𝑥 5
3
− 𝑥𝑥 4
1
2
4. 𝑓𝑓 ′ (𝑥𝑥) = 3 sec 2 𝑥𝑥 − 5 sec 𝑥𝑥 tan 𝑥𝑥 − 10 sin 𝑥𝑥 − √1−𝑥𝑥 2
5. 𝑔𝑔′(𝑥𝑥) =
1
𝑥𝑥
�𝑒𝑒 𝑥𝑥 + ��𝑥𝑥 2 +1�−2𝑥𝑥(𝑒𝑒 𝑥𝑥 +ln 𝑥𝑥)
(𝑥𝑥 2 +1)2
2
(do not simplify answer)
6. ℎ′ (𝑥𝑥) = tan 𝑥𝑥 + 𝑥𝑥 sec 𝑥𝑥 − sin 𝑥𝑥
2𝑥𝑥
4
7. 𝑦𝑦 ′ = 12(3𝑥𝑥 − 1)3 + 5𝑒𝑒 5𝑥𝑥 − 𝑥𝑥 2 +1 − 1+(4𝑥𝑥+3)2
2
2
2
8. 𝑓𝑓 ′ (𝑡𝑡) = 5(2𝑡𝑡𝑒𝑒 𝑡𝑡 + sin 𝑡𝑡)sin4�𝑒𝑒 𝑡𝑡 − cos 𝑡𝑡� cos�𝑒𝑒 𝑡𝑡 − cos 𝑡𝑡�
9. 𝑦𝑦 ′ =
−1
𝑥𝑥 2
(𝑥𝑥+1)2 �1−�
�
𝑥𝑥+1
1
− 𝑒𝑒 −𝑥𝑥 csc 𝑥𝑥 − 𝑒𝑒 −𝑥𝑥 csc 𝑥𝑥 cot 𝑥𝑥 (could be simplified more, if you’re brave!)
10. 𝑓𝑓 ′ (𝑥𝑥) = 1 + (2 ln 3)𝑥𝑥 − 7sec 𝑥𝑥 (ln 7) sec 𝑥𝑥 tan 𝑥𝑥
𝑑𝑑𝑑𝑑
2𝑥𝑥+2𝑦𝑦
11. 𝑑𝑑𝑑𝑑 = 6𝑦𝑦−2𝑥𝑥
𝑑𝑑𝑑𝑑
12. 𝑑𝑑𝑑𝑑 =
10𝑥𝑥𝑥𝑥+𝑦𝑦
𝑥𝑥
13. 𝑦𝑦 = 54𝑥𝑥 − 27
14. 𝑦𝑦 = −2, 𝑦𝑦 = 𝑥𝑥 + 2
15. 𝑥𝑥 = ln �1/2
16. 𝑥𝑥 = ±�4⁄5
17.
18.