Derivadas

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Excel Man 
Aula de Cálculos I e Física Teórica 
https://www.youtube.com/channel/UCEA5h
d2pxzQPpkpsAPPnWtQ 
Derivadas 
 
1) y=k y’ =0 
2) y = ax y’ =a 
3) y=x
n 
 y’ = nx
n-1
 
4) y=u
n 
 y’ = nu
n-1
u’ 
5) y=u.v y’ = u.v’ + v.u’ 
6)y= �� y’=
�.����.��
��	
 
7) y =e
x 
y’ = e
x
 
8) y =e
u 
y’ = e
u
.u’ 
9)y=a
x
 y´=a
x
.ln(a) 
10)y=a
u
 y’=a
u
ln(a).u’ 
11)y=ln(u) y’=
��
� 
12)y=sen u y’=cos u.u’ 
13)y=cos u y’=-sen u.u’ 
14)y=tg u y’=sec
2
u.u’ 
15)� = �� = �´ = ������´ + ln��	 , �´ 
 
 
 
1 – �5� + 12	��8� − 7	� 
u.v’ + v.u’ u=�5� + 12	� v=�8� − 7	� 
nu
n-1
u’ 
 u’=3�5� + 12	�. 5 v’=4�8� − 7	�. 8 
 �5� + 12	�. 4�8� − 7	�. 8 + �8� − 7	�. 3�5� + 12	�. 5 
 
32. �5� + 12	�. �8� − 7	� + 15. �8� − 7	�. �5� + 12	� 
 
2-!��	 = �"#$�	
�"#��	% 
�.����.��
��	
 
 
� = �5� + 3	� � = �5� − 3	� 
nu
n-1
u’ 
�′ = 2�5� + 3	�. 5 + 0 �′ = 3�5� − 3	�. 5 − 0 
�5� − 3	3. 2�5� + 3	.5 − �5� + 3	2. 3�5� − 3	2. 5
(�5� − 3	3)�
 
10. �5� − 3	3. �5� + 3	 − 15. �5� + 3	2. �5� − 3	2
�5� − 3	* 
 
 
 
 
 
 
 
3- !��	 = +,-���. ./+��� y=u.v y’ = u.v’ + v.u’ 
 
� = +,-��� � = ./+��� 
nu
n-1
u’ 
�′ = 3+,-�����. ./+��. 2� 
�′ = 2./+�����. �−+,-��	. 3�� 
+,-���. 2./+�����. �−+,-��	. 3�� + ./+���. 3+,-�����. ./+��. 2� 
 
4- !��	 = 3,�*#0$�#%$�#
	 
y =e
u 
� = 6�� + 3�� + 2�� 
y=x
n 
 y’ = nx
n-1
 
�′ = 6. 4���� + 3. 3���� + 2.2���� 
�′ = 24�� + 9�� + 4� 
!��#	 = 3,(*#0$�#%$�#
). 24�� + 9�� + 4� 
 
 
 
 
 
 
 
 
 
5 - !�3	 = 4567
√9
 
y=a
u
 y’=a
u
ln(a).u’ 
� = √3 = 3
:
 �� = �� 3
:
�� 
�� = 12 3
��� 
!′�3	 = 4567
√9
. ln 4567. 
�
� 3
�:
 
!′�3	 = 4567
√9
. ln 4567. 
�
�√9 
 
� = �� = �´ = ������´ + ln��	 , �´ 
6 - !��	 = �4� + 2	("#%$�) 
� = 4� + 2 �′ = 4 
� = 5�� + 3 �′ = 5.3���� + 0 �′ = 15�� 
!´��	 = �5�� + 3	. �4� + 2	"#%$�. 4 + ln�4� + 2	 . 15�� 
 
 
 
7 - !��	 = ;/<��5�� + 6��	 ;/<5� = �´� . ;/<5, 
� = 5�� + 6�� �� = 5.3���� + 6.4���� 
�� = 15�� + 24�� 
!´��	 = 15�
� + 24��
5�� + 6�� . ;/<�, 
 
8 - !��	 = ln �=�
:
 + >�
%
) 
!��	 = ln �=√� + >√��) 
y=ln(u) y’=
��
� 
� = =�
�
� + >�
�
� 
�′ = = 12 �
�
��� + > 32 �
�
��� 
�′ = = 12 �
��� + > 32 �
�
� 
!′��	 =
= 12 �
��� + > 32 �
�
�
=�
�
� + >�
�
�
 
!′��	 =
= 12√? + >
3√?
2
=√� + >√��
 
 
9 – !��	 = �# $ "�# 
�. �� − �. �′
��	� 
U=2x+5 u’=2 
V=4x v’=4 
!���	 = �. �
� − �. �′
��	� 
!���	 = 4�. 2 − �2� + 5	.4�4�	� 
!���	 = 8� − 8� − 20�4�	� 
!���	 = −20�4�	� 
!���	 = −204�. 4� 
!���	 = −2016�� 
!���	 = −54�� 
 
10 -!��	 = �2� + 1	. �3� − 2	 y=u.v y’ = u.v’ + v.u’ 
� = 2� + 1 � = 3� − 2 
�′ = 2 �′ = 3 
!���	 = �. �� + �. �′ 
!���	 = �2� + 1	.3 + �3� − 2	.2 
!′��	 = �6� + 3	 + �6� − 4	 
!′��	 = 12� − 1 
 
 
11 - � = 10. �3� + 1	. �1 − 5�	 
� = �30� + 10	. �1 − 5�	 
� = 30� + 10 
�′ = 30 
� = 1 − 5� 
�� = −5 
!���	 = �. �� + �. � 
!���	 = �30� + 10	. −5 + �1 − 5�	.30 
!���	 = −150� − 50 + 30 − 150�	 
!���	 = −300� − 20 
 
 
 
12 - !��	 = �� . ��
" − 2�� + 1	 
� = 12 
�� = 0 
� = �" − 2�� + 1 
�′ = 5�� − 6�� 
!���	 = �. �� + �. �′ 
!���	 = 12 . �5�
� − 6��	 + ��" − 2�� + 1	.0 
!���	 = 12 . �5�
� − 6��	 + 0 
!���	 = ��.( 5�
� − 6��) 
 
13 - 
#$�
#�� 
!���	 = �. �
� − �. �′
��	� 
� = � + 1 
�′ = 1 
� = � − 2 
�′ = 1 
!���	 = � − 2 − � − 1�� − 2	� 
!���	 = −2 − 1�� − 2	� 
!���	 = −3�� − 2	�