DIFFERENTIATION
−𝟏 𝒕
, 𝒚 = √𝒂𝐜𝐨𝐬
−𝟏 𝒕
1
If 𝒙 = √𝒂𝐬𝐢𝐧
2
If 𝒙 = 𝐜𝐨𝐬 𝒕 + 𝒕 𝐬𝐢𝐧 𝒕 , 𝒚 = 𝐬𝐢𝐧 𝒕 − 𝒕 𝐜𝐨𝐬 𝒕 , show that
3
If 𝒙 = 𝒂 𝐜𝐨𝐬 𝒕 + 𝒃 𝐬𝐢𝐧 𝒕 , 𝒚 = 𝒂 𝐬𝐢𝐧 𝒕 − 𝒃 𝐜𝐨𝐬 𝒕 , prove that 𝒚𝟐
4
If 𝒙 = 𝒂(𝟏 − 𝐜𝐨𝐬 𝜽) , 𝒚 = 𝒂(𝜽 + 𝐬𝐢𝐧 𝜽) 𝒑𝒓𝒐𝒗𝒆 𝒕𝒉𝒂𝒕
5
If 𝒙 = 𝐬𝐢𝐧−𝟏 (𝟏+𝒕𝟐 ) , 𝒚 = 𝐭𝐚𝐧−𝟏 (𝟏−𝒕𝟐 ) , 𝒕 > 1 , 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡
6
If 𝒚 = 𝑨𝒆𝒎𝒙 + 𝑩𝒆𝒏𝒙 ,prove that
7
If 𝒚 = 𝒆𝒂 𝐜𝐨𝐬
8
If
9
𝒂 > 0 , −1 < 𝑡 < 1 show that
𝟐𝒕
𝒙𝟐
𝒂𝟐
−𝟏 𝒙
𝒅𝟐 𝒚
𝒅𝒙𝟐
=
𝒅𝟐 𝒚
− (𝒎 + 𝒏)
𝒅𝒙𝟐
𝒅𝒚
𝒅𝒙
=
−𝒚
𝒙
𝒔𝒆𝒄𝟑 𝒕
𝒕
𝒅𝟐 𝒚
𝒅𝒙𝟐
−𝒙
−𝟏
𝒅𝒚
𝒅𝒙
+𝒚 = 𝟎
𝜽
𝜽
= 𝟒𝒂 𝐬𝐞𝐜 (𝟐) 𝒄𝒐𝒔𝒆𝒄𝟑 (𝟐)
𝟐𝒕
𝒅𝒚
𝒅𝒙
= −𝟏
+ 𝒎𝒏𝒚 = 𝟎
𝒑𝒓𝒐𝒗𝒆 𝒕𝒉𝒂𝒕 (𝟏 − 𝒙𝟐 )𝒚𝟐 − 𝒙𝒚𝟏 − 𝒂𝟐 𝒚 = 𝟎
𝒚𝟐
𝒅𝟐 𝒚
𝒅𝒙𝟐
+ 𝒃𝟐 = 𝟏 , prove that
−𝒃𝟒
= 𝒂𝟐 𝒚𝟑
If y = 𝐭𝐚𝐧 𝒙 + 𝐬𝐞𝐜 𝒙 , prove that (𝟏 − 𝐬𝐢𝐧 𝒙)𝟐
𝟐
10
If y = [𝒍𝒐𝒈 (𝒙 + √𝒙𝟐 + 𝟏)]
11
Prove that 𝒙𝟑 𝒚𝟐 = (𝒙 𝒚𝟏 − 𝒚)𝟐 , if ( a+ bx ) 𝒆
𝒅𝟐 𝒚
𝒅𝒙𝟐
= 𝐜𝐨𝐬 𝒙 .
then prove that , ( 1 + 𝒙𝟐 )𝒚𝟐 + 𝒙 𝒚𝟏 = 𝟐
𝒚⁄
𝒙
=𝒙
𝒅𝟐 𝒚
12 y = 𝒆𝒙 𝐭𝐚𝐧−𝟏 𝒙 , prove that ( 1+𝒙𝟐 )𝒅𝒙𝟐 − 𝟐(𝟏 − 𝒙 + 𝒙𝟐 )
𝒅𝟐 𝒚
𝒅𝟐 𝒚
𝒅𝒙𝟐
𝒅𝒚
𝒅𝒙
𝟏
(𝟏 − 𝒙)𝟐 𝒚 = 𝟎
𝒕
13
Find
14
If 𝒚𝟑 + 𝒙𝟑 − 𝟑𝒂𝒙𝒚 = 𝟎 , prove that
15
If 𝒙 = 𝐬𝐞𝐜 𝜽 − 𝐜𝐨𝐬 𝜽 , 𝒚 = 𝒔𝒆𝒄𝒏 𝜽 − 𝒄𝒐𝒔𝒏 𝜽 ,prove that (𝒙𝟐 + 𝟒)(𝒚| ) = 𝒏𝟐 (𝒚𝟐 + 𝟒)
𝒅𝒙𝟐
, given that , x = a [𝐜𝐨𝐬 𝒕 +
𝒅𝒚
+
𝒅𝒙
𝟐
𝒅𝟐 𝒚
𝒅𝒙𝟐
𝒍𝒐𝒈 𝒕𝒂𝒏𝟐 ( )] and y = a 𝐬𝐢𝐧 𝒕
𝟐
−𝟐𝒂𝟑 𝒙𝒚
= (𝒚𝟐 −𝒂𝒙)𝟑
𝟐
16 If 𝒙 = 𝟐 𝐜𝐨𝐬 𝜽 − 𝐜𝐨𝐬 𝟐𝜽 , 𝒚 = 𝟐 𝐬𝐢𝐧 𝜽 − 𝐬𝐢𝐧 𝟐𝜽 𝒕𝒉𝒆𝒏
17
If 𝒚 = [√𝒙 + 𝟏 + √𝒙 − 𝟏]
𝒎
𝒅𝒚
𝒅𝒙
= 𝐭𝐚𝐧(𝒑𝜽) , 𝒇𝒊𝒏𝒅 𝒑
𝟏
, prove that (𝒙𝟐 − 𝟏)𝒚𝟐 + 𝒙𝒚𝟏 = 𝟒 𝒎𝟐 𝒚
18
If 𝒇(𝒙) = |𝒙|𝟑 show that 𝒇|| (𝒙) exist for all real , then find 𝒇|| (𝒙) .
19
If 𝒚 = 𝒙 +
𝟏
𝒙+
𝒙+
𝟏
prove that
𝟏
𝟏
𝒙+𝒙 ++++++++∞
20
Prove that the derivative of 𝐭𝐚𝐧−𝟏 (
21∗
If 𝒚 = 𝐬𝐢𝐧−𝟏
𝟏
+ 𝐭𝐚𝐧−𝟏
𝟐
√𝟏+𝒙
√𝟏+𝒙𝟐 −𝟏
√𝟏+𝒙𝟐 −𝟏
𝒙
𝒅𝒚
𝒅𝒙
𝒙
𝒚
= 𝟐𝒚−𝒙
) 𝒘𝒊𝒕𝒉 𝒓𝒆𝒔𝒑𝒆𝒄𝒕 𝒕𝒐 𝐭𝐚𝐧−𝟏 (
, then show that
𝒅𝒚
𝒅𝒙
−𝟏
= 𝟐(𝟏+𝒙𝟐 )
𝟐𝒙√𝟏−𝒙𝟐
)
𝟏−𝟐𝒙𝟐
𝟏
𝒂𝒕 𝒙 = 𝟎 𝒊𝒔 𝟒
𝒅𝒚
22
If 𝒙 = 𝒂 𝐬𝐢𝐧 𝟐𝜽(𝟏 + 𝐜𝐨𝐬 𝟐𝜽) 𝒂𝒏𝒅 𝒚 = 𝒃 𝐜𝐨𝐬 𝟐𝜽(𝟏 − 𝐜𝐨𝐬 𝟐𝜽) , find 𝒅𝒙
23
Find
24
If 𝒙 =
25
If 𝒙 = 𝒆
26∗
if 𝒚 = 𝐬𝐢𝐧−𝟏 (𝟏+(𝟑𝟔)𝒙 ) prove that
27
If 𝒚 = 𝒍𝒐𝒈(𝟏 + 𝐜𝐨𝐬 𝒙) ,prove that
28
If 𝒚 = (𝐬𝐢𝐧−𝟏 𝒙) + (𝐜𝐨𝐬−𝟏 𝒙) , prove that (𝟏 − 𝒙𝟐 ) 𝒅𝒙𝟐 − 𝒙 𝒅𝒙 = 𝟒
29∗
If 𝒚 =
30
If 𝒙 = 𝒆𝒚 , prove that
31
Find the derivative of :
32
If 𝒚 =
33
If 𝒙 = 𝒂(𝐜𝐨𝐬 𝟐𝒕 + 𝟐𝒕 𝐬𝐢𝐧 𝟐𝒕) 𝒂𝒏𝒅 𝒚 = 𝒂(𝐬𝐢𝐧 𝟐𝒕 − 𝟐𝒕 𝐜𝐨𝐬 𝟐𝒕) , then find 𝒅𝒙𝟐 .
34
If 𝒙 = 𝒂 𝐬𝐢𝐧 𝒑𝒕 , 𝒚 = 𝒃 𝐜𝐨𝐬 𝒑𝒕 , show that (𝒂𝟐 − 𝒙𝟐 )𝒚
𝒅𝒚
𝒅𝒙
𝟏 𝒂
𝟏
, 𝒚 = 𝒂(𝒕+ 𝒕 ) , 𝒂 > 0
, when 𝒙 = (𝒕 + 𝒕 )
𝟏+𝒍𝒐𝒈𝒕
𝒕𝟐
𝐭𝐚𝐧−𝟏 (
𝟑+𝟐𝒍𝒐𝒈𝒕
𝒕
𝒚=
𝒚−𝒙𝟐
)
𝒙𝟐
, prove that
𝒅𝒚
𝒅𝒙
𝟐𝒙+𝟏 𝟑𝒙
𝟐
𝟐
√𝟓
𝐭𝐚𝐧−𝟏 (
𝟏
𝒙
𝒅𝒚
𝒅𝒙
=
𝟐 .𝟔𝒙 𝒍𝒐𝒈𝟔
𝟏+(𝟑𝟔)𝒙
𝒅𝟑 𝒚
𝒅𝟐 𝒚
+
𝟑
𝒅𝒙
𝒅𝒙𝟐
𝒙
, prove that
𝒅𝒚
𝒅𝒙
𝒅𝒚
. 𝒅𝒙 = 𝟎
𝒅𝟐 𝒚
𝐭𝐚𝐧 𝟐) , 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕
𝒙
𝟏
= 𝒙[𝒔𝒆𝒄𝟐 (𝒍𝒐𝒈𝒙) + 𝟐 + 𝟐 𝐭𝐚𝐧(𝒍𝒐𝒈𝒙)]
𝟐
√𝟓
𝒅𝒚 𝟐
𝒅𝒚
, 𝒕 > 0 then show that , 𝒚 𝒅𝒙 = 𝟐𝒙 (𝒅𝒙) + 𝟏
𝒅𝒚
𝒅𝒙
=
𝒅𝒚
𝟏
𝟑+𝟐 𝐜𝐨𝐬 𝒙
𝒙−𝒚
= 𝒙𝒍𝒐𝒈𝒙
𝐬𝐢𝐧(𝒙𝒙 ) + (𝐬𝐢𝐧 𝒙)𝒙
𝒅𝒚
√𝟏+𝒚𝟒
+
𝒅𝒙
√𝟏+𝒙𝟒
=𝟎
𝒅𝟐 𝒚
𝒅𝟐 𝒚
𝒅𝒙𝟐
+ 𝒃𝟐 = 𝟎 .