Differentiation Rules for June Test 2016 (Class Test 2)
𝑑𝑑
(𝑐𝑐) = 0, 𝑐𝑐 constant
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑑𝑑
(𝑐𝑐𝑐𝑐(𝑥𝑥)) = 𝑐𝑐 �𝑓𝑓(𝑥𝑥)� = 𝑐𝑐𝑓𝑓′(𝑥𝑥)
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
𝑑𝑑
[𝑓𝑓(𝑥𝑥) ± 𝑔𝑔(𝑥𝑥)] = 𝑓𝑓 ′ (𝑥𝑥) ± 𝑔𝑔′ (𝑥𝑥)
𝑑𝑑𝑑𝑑
𝑑𝑑
[𝑓𝑓(𝑥𝑥)𝑔𝑔(𝑥𝑥)] = 𝑓𝑓 ′ (𝑥𝑥)𝑔𝑔(𝑥𝑥) + 𝑓𝑓(𝑥𝑥)𝑔𝑔′ (𝑥𝑥) (Product Rule)
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑓𝑓(𝑥𝑥) 𝑓𝑓 ′ (𝑥𝑥)𝑔𝑔(𝑥𝑥) − 𝑓𝑓(𝑥𝑥)𝑔𝑔′ (𝑥𝑥)
� �= (Quotient Rule)
𝑑𝑑𝑑𝑑 𝑔𝑔(𝑥𝑥) [𝑔𝑔(𝑥𝑥)]2
𝑑𝑑
(𝑥𝑥 𝑛𝑛 ) = 𝑛𝑛𝑥𝑥 𝑛𝑛−1 (Power Rule)
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑥𝑥
(𝑒𝑒 ) = 𝑒𝑒 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑥𝑥
(𝑎𝑎 ) = 𝑎𝑎 𝑥𝑥 ln 𝑎𝑎
𝑑𝑑𝑑𝑑
𝑑𝑑 1
ln 𝑥𝑥 = , 𝑥𝑥 ∈ ℝ, 𝑥𝑥 > 0
𝑑𝑑𝑑𝑑 𝑥𝑥
𝑑𝑑 1
ln|𝑥𝑥| = , 𝑥𝑥 ∈ ℝ
𝑑𝑑𝑑𝑑 𝑥𝑥
𝑑𝑑 1
(log 𝑎𝑎 𝑥𝑥) =
𝑑𝑑𝑑𝑑 𝑥𝑥 ln 𝑎𝑎
𝑑𝑑
(sin 𝑥𝑥) = cos 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(cos 𝑥𝑥) = − sin 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(tan 𝑥𝑥) = sec 2 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(cot 𝑥𝑥) = − csc 2 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(sec 𝑥𝑥) = sec 𝑥𝑥 tan 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(csc 𝑥𝑥) = − csc 𝑥𝑥 cot 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑 1
(arcsin 𝑥𝑥) =
𝑑𝑑𝑑𝑑 √1 − 𝑥𝑥 2
𝑑𝑑
(𝑐𝑐) = 0, 𝑐𝑐 constant
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑑𝑑
(𝑐𝑐𝑐𝑐(𝑥𝑥)) = 𝑐𝑐 �𝑓𝑓(𝑥𝑥)� = 𝑐𝑐𝑓𝑓′(𝑥𝑥)
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
𝑑𝑑
[𝑓𝑓(𝑥𝑥) ± 𝑔𝑔(𝑥𝑥)] = 𝑓𝑓 ′ (𝑥𝑥) ± 𝑔𝑔′ (𝑥𝑥)
𝑑𝑑𝑑𝑑
𝑑𝑑
[𝑓𝑓(𝑥𝑥)𝑔𝑔(𝑥𝑥)] = 𝑓𝑓 ′ (𝑥𝑥)𝑔𝑔(𝑥𝑥) + 𝑓𝑓(𝑥𝑥)𝑔𝑔′ (𝑥𝑥) (Product Rule)
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑓𝑓(𝑥𝑥) 𝑓𝑓 ′ (𝑥𝑥)𝑔𝑔(𝑥𝑥) − 𝑓𝑓(𝑥𝑥)𝑔𝑔′ (𝑥𝑥)
� �= (Quotient Rule)
𝑑𝑑𝑑𝑑 𝑔𝑔(𝑥𝑥) [𝑔𝑔(𝑥𝑥)]2
𝑑𝑑
(𝑥𝑥 𝑛𝑛 ) = 𝑛𝑛𝑥𝑥 𝑛𝑛−1 (Power Rule)
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑥𝑥
(𝑒𝑒 ) = 𝑒𝑒 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑥𝑥
(𝑎𝑎 ) = 𝑎𝑎 𝑥𝑥 ln 𝑎𝑎
𝑑𝑑𝑑𝑑
𝑑𝑑 1
ln 𝑥𝑥 = , 𝑥𝑥 ∈ ℝ, 𝑥𝑥 > 0
𝑑𝑑𝑑𝑑 𝑥𝑥
𝑑𝑑 1
ln|𝑥𝑥| = , 𝑥𝑥 ∈ ℝ
𝑑𝑑𝑑𝑑 𝑥𝑥
𝑑𝑑 1
(log 𝑎𝑎 𝑥𝑥) =
𝑑𝑑𝑑𝑑 𝑥𝑥 ln 𝑎𝑎
𝑑𝑑
(sin 𝑥𝑥) = cos 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(cos 𝑥𝑥) = − sin 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(tan 𝑥𝑥) = sec 2 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(cot 𝑥𝑥) = − csc 2 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(sec 𝑥𝑥) = sec 𝑥𝑥 tan 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑
(csc 𝑥𝑥) = − csc 𝑥𝑥 cot 𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑 1
(arcsin 𝑥𝑥) =
𝑑𝑑𝑑𝑑 √1 − 𝑥𝑥 2
