, XII- MATHS Derivatives- Results
𝑑
I. (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1 [Algebraic expressions : Variable constant ]
𝑑𝑥
𝑑 𝑑 𝑑
(𝑖) (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) = 0 (ii) 𝑑𝑥 (𝑐) = 0 (iii) 𝑑𝑥 (𝑥) = 1
𝑑𝑥
𝑑 𝑑 𝑑
(iv) 𝑑𝑥 (𝑥 2 ) = 2𝑥 (v) 𝑑𝑥 (𝑥 3 ) = 3𝑥 2 (vi) (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1
𝑑𝑥
𝑑 1 −1 𝑑 1
(vii) 𝑑𝑥 (𝑥) = (viii) 𝑑𝑥 (√𝑥) = 2
𝑥2 √𝑥
An algebraic expression can be differentiated whenever it is in power/ index form,
𝟏
𝒏 𝟏
if not, change to power form. Remember √𝒙 = 𝒙𝒏 , = 𝒙−𝒏
𝒙𝒏
𝑑 𝑑
II. (sin 𝑥) = 𝑐𝑜𝑠𝑥 (𝐜𝐨s 𝑥) = −𝑠𝑖𝑛𝑥 Derivative of trigonometric
𝑑𝑥 𝑑𝑥
function which starts with
𝑑 𝑑
(𝑡𝑎𝑛𝑥) = 𝑠𝑒𝑐 2 𝑥 (𝒄𝒐𝑡𝑥) = −𝑐𝑜𝑠𝑒𝑐 2 𝑥 ‘’Co ‘’ will have a negative
𝑑𝑥 𝑑𝑥
sign.
𝑑 𝑑
(𝑠𝑒𝑐𝑥) = 𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥 (𝐜𝐨sec 𝑥) = −𝑐𝑜𝑠𝑒𝑐𝑥 𝑐𝑜𝑡𝑥
𝑑𝑥 𝑑𝑥
III.
𝑑 1 𝑑 −1
(sin−1 𝑥) = (𝐜𝐨s −1 𝑥) =
𝑑𝑥 √1−𝑥 2 𝑑𝑥 √1−𝑥 2
𝑑 1 𝑑 −1
(tan−1 𝑥) = (𝐜𝐨t −1 𝑥) =
𝑑𝑥 1+𝑥 2 𝑑𝑥 1+𝑥 2
𝑑 1 𝑑 −1
(sec −1 𝑥) = (𝐜𝐨sec −1 𝑥) =
𝑑𝑥 𝑥√𝑥 2 −1 𝑑𝑥 𝑥√𝑥 2 −1
[Power form : Constantvariable ]
𝑑 𝑑
IV. (𝑒 𝑥 ) = 𝑒 𝑥 (𝑎 𝑥 ) = 𝑎 𝑥 log 𝑎 All log functions are with base ‘e’
𝑑𝑥 𝑑𝑥
𝑑 1 𝑑 1 unless otherwise specified.
V. (𝑙𝑜𝑔𝑥) = (𝑙𝑜𝑔𝑎 𝑥) = ,
𝑑𝑥 𝑥 𝑑𝑥 𝑥𝑙𝑜𝑔𝑎
𝑑 𝑑
VI. 𝑘𝑓(𝑥) = 𝑘 𝑑𝑥 𝑓(𝑥)
𝑑𝑥
𝑑 𝑑 𝑑
(𝑓(𝑥) ± 𝑔(𝑥)) = 𝑓(𝑥) ± 𝑔(𝑥)
𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑑 𝑑 𝑑 𝒅
(𝑓(𝑥)𝑔(𝑥)) = 𝑓(𝑥) 𝑔(𝑥) + 𝑔(𝑥) 𝑑𝑥 𝑓(𝑥) OR (𝒖𝒗) = 𝒖 𝒗′ + 𝒗𝒖′ [Product Rule]
𝑑𝑥 𝑑𝑥 𝒅𝒙
𝒅 𝒖 𝒗 𝒖′ −𝒖𝒗′
( )= [Quotient Rule]
𝒅𝒙 𝒗 𝒗𝟐
VII. Chain Rule / Function of function Rule
𝑑 𝒅
(𝒇(𝒈(𝒙)) = 𝒇′ (𝒈(𝒙)) (𝒈(𝒙)) = 𝒇′ (𝒈(𝒙))𝒈′ (𝒙).
𝑑𝑥 𝒅𝒙
𝑑 𝒅
(𝒇(𝒈(𝒉(𝒙))) = 𝒇′(𝒈(𝒉(𝒙))𝒈′(𝒉(𝒙)) (𝒉(𝒙)) = 𝒇′(𝒈(𝒉(𝒙))𝒈′(𝒉(𝒙))(𝒉′(𝒙))
𝑑𝑥 𝒅𝒙
© Dineshmaths 050 9312 916 XII Mathematics
𝑑
I. (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1 [Algebraic expressions : Variable constant ]
𝑑𝑥
𝑑 𝑑 𝑑
(𝑖) (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) = 0 (ii) 𝑑𝑥 (𝑐) = 0 (iii) 𝑑𝑥 (𝑥) = 1
𝑑𝑥
𝑑 𝑑 𝑑
(iv) 𝑑𝑥 (𝑥 2 ) = 2𝑥 (v) 𝑑𝑥 (𝑥 3 ) = 3𝑥 2 (vi) (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1
𝑑𝑥
𝑑 1 −1 𝑑 1
(vii) 𝑑𝑥 (𝑥) = (viii) 𝑑𝑥 (√𝑥) = 2
𝑥2 √𝑥
An algebraic expression can be differentiated whenever it is in power/ index form,
𝟏
𝒏 𝟏
if not, change to power form. Remember √𝒙 = 𝒙𝒏 , = 𝒙−𝒏
𝒙𝒏
𝑑 𝑑
II. (sin 𝑥) = 𝑐𝑜𝑠𝑥 (𝐜𝐨s 𝑥) = −𝑠𝑖𝑛𝑥 Derivative of trigonometric
𝑑𝑥 𝑑𝑥
function which starts with
𝑑 𝑑
(𝑡𝑎𝑛𝑥) = 𝑠𝑒𝑐 2 𝑥 (𝒄𝒐𝑡𝑥) = −𝑐𝑜𝑠𝑒𝑐 2 𝑥 ‘’Co ‘’ will have a negative
𝑑𝑥 𝑑𝑥
sign.
𝑑 𝑑
(𝑠𝑒𝑐𝑥) = 𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥 (𝐜𝐨sec 𝑥) = −𝑐𝑜𝑠𝑒𝑐𝑥 𝑐𝑜𝑡𝑥
𝑑𝑥 𝑑𝑥
III.
𝑑 1 𝑑 −1
(sin−1 𝑥) = (𝐜𝐨s −1 𝑥) =
𝑑𝑥 √1−𝑥 2 𝑑𝑥 √1−𝑥 2
𝑑 1 𝑑 −1
(tan−1 𝑥) = (𝐜𝐨t −1 𝑥) =
𝑑𝑥 1+𝑥 2 𝑑𝑥 1+𝑥 2
𝑑 1 𝑑 −1
(sec −1 𝑥) = (𝐜𝐨sec −1 𝑥) =
𝑑𝑥 𝑥√𝑥 2 −1 𝑑𝑥 𝑥√𝑥 2 −1
[Power form : Constantvariable ]
𝑑 𝑑
IV. (𝑒 𝑥 ) = 𝑒 𝑥 (𝑎 𝑥 ) = 𝑎 𝑥 log 𝑎 All log functions are with base ‘e’
𝑑𝑥 𝑑𝑥
𝑑 1 𝑑 1 unless otherwise specified.
V. (𝑙𝑜𝑔𝑥) = (𝑙𝑜𝑔𝑎 𝑥) = ,
𝑑𝑥 𝑥 𝑑𝑥 𝑥𝑙𝑜𝑔𝑎
𝑑 𝑑
VI. 𝑘𝑓(𝑥) = 𝑘 𝑑𝑥 𝑓(𝑥)
𝑑𝑥
𝑑 𝑑 𝑑
(𝑓(𝑥) ± 𝑔(𝑥)) = 𝑓(𝑥) ± 𝑔(𝑥)
𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑑 𝑑 𝑑 𝒅
(𝑓(𝑥)𝑔(𝑥)) = 𝑓(𝑥) 𝑔(𝑥) + 𝑔(𝑥) 𝑑𝑥 𝑓(𝑥) OR (𝒖𝒗) = 𝒖 𝒗′ + 𝒗𝒖′ [Product Rule]
𝑑𝑥 𝑑𝑥 𝒅𝒙
𝒅 𝒖 𝒗 𝒖′ −𝒖𝒗′
( )= [Quotient Rule]
𝒅𝒙 𝒗 𝒗𝟐
VII. Chain Rule / Function of function Rule
𝑑 𝒅
(𝒇(𝒈(𝒙)) = 𝒇′ (𝒈(𝒙)) (𝒈(𝒙)) = 𝒇′ (𝒈(𝒙))𝒈′ (𝒙).
𝑑𝑥 𝒅𝒙
𝑑 𝒅
(𝒇(𝒈(𝒉(𝒙))) = 𝒇′(𝒈(𝒉(𝒙))𝒈′(𝒉(𝒙)) (𝒉(𝒙)) = 𝒇′(𝒈(𝒉(𝒙))𝒈′(𝒉(𝒙))(𝒉′(𝒙))
𝑑𝑥 𝒅𝒙
© Dineshmaths 050 9312 916 XII Mathematics
