Conclusion 𝑓 𝑥 = 𝑥 derivative 𝑓′ 𝑥 = 1𝑥 =𝑥 =1 𝑓 𝑥 = 𝑥 derivative 𝑓′(𝑥) = 2𝑥 = 2𝑥 = 2𝑥 𝑓 𝑥 = 𝑥 derivative 𝑓′(𝑥) = 3𝑥 = 3𝑥 𝑓 𝑥 = 𝑥 derivative 𝑓′(𝑥) = 4𝑥 = 4𝑥 5. 𝑓 𝑥 = 𝑥 derivative 𝑓′(𝑥) = 5𝑥 = 5𝑥 1. 2. 3. 4. 6. 𝑓 𝑥 = = 𝑥 derivative 𝑓′(𝑥) = −1𝑥 7. 𝑓 𝑥 = 𝑥= 𝑥 derivative 𝑓′(𝑥) = 𝑥 = −𝑥 = 𝑥 =− = = Observing 𝑓 𝑥 = 𝑥 derivative 𝑓′(𝑥) = 𝑛𝑥 Note Hum ne 1 to 5 functions ka derivative nakala he, tu 1 to 5 questions se ye andaza nekla k hum har functions ka derivative nekal sakte he, tu hum further function ko solve nahi karte he bus unka derivative yaad kar lete he or us concept ko hum derivative of simple function kahe ge Simple function This is Simple function of algebraic, Exponential and logarithmic and its derivative Note Derivative commonly notation is or S# Functions Derivative 1 𝒚=𝒄 2 𝒚 = 𝒙𝒏 3 𝒚 = 𝒆𝒙 4 𝒇 𝒙 = 𝒂𝒙 𝒅𝒚 =𝟎 𝒅𝒙 𝒅𝒚 = 𝒏𝒙𝒏 𝟏 𝒅𝒙 𝒅𝒚 = 𝒆𝒙 𝒅𝒙 𝒇′ 𝒙 = 𝒂𝒙 𝒍𝒐𝒈𝒂 5 𝒇 𝒙 = 𝒍𝒏𝒙 6 𝒇 𝒙 = 𝒍𝒐𝒈𝒂 𝒙 𝟏 𝒙 𝟏 𝒇′ 𝒙 = 𝒙𝒍𝒐𝒈𝒂 𝒇′ 𝒙 = Examples derivative derivative derivative derivative derivative derivative 7) derivative Home work 𝒙 𝒙 𝒂 𝒙 𝒙 𝒙 𝒙 derivative derivative derivative derivative derivative derivative 𝒂 derivative Simple function This is Simple function of trigonometric and its derivative Note Derivative commonly notation is or S# Functions Derivative 1 2 3 4 5 6 𝟐 𝟐 Examples 𝑦 = 𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙 find derivative? Solution: 1. Given, 𝑦 = 𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙 Taking derivative with respect to x 𝑑𝑦 𝑑 𝑑 = 𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙 𝑑𝑥 𝑑𝑥 𝑑𝑥 Using formula of derivative, such as = 𝒄𝒐𝒔𝒙 − 𝒔𝒊𝒏𝒙 (Ans) 2) 𝑦 = 𝒕𝒂𝒏𝒙 + 𝒄𝒐𝒕𝒙 find derivative? Solution: Given, 𝑦 = 𝒕𝒂𝒏𝒙 + 𝒄𝒐𝒕𝒙 Taking derivative with respect to x 𝑑𝑦 𝑑 𝑑 = 𝒕𝒂𝒏𝒙 + 𝒄𝒐𝒕𝒙 𝑑𝑥 𝑑𝑥 𝑑𝑥 Using formula of derivative, such as = 𝐬𝐞𝐜 𝟐 𝒙 − 𝒄𝒐𝐬𝐞𝐜 𝟐 𝒙(Ans) 3) Solution: find derivative? Given, Taking derivative with respect to x Using formula of derivative, such as (Ans) 4) Solution: Taking derivative with respect to x 𝟐 + 𝟐 find derivative? + (Ans) Home work 𝒙 derivative derivative derivative derivative derivative Simple function This is Simple function of trigonometric and its derivative Note Derivative commonly notation is or S# 1 2 Functions 𝒇 𝒙 = 𝐬𝐢𝐧 𝟏 𝒇 𝒙 = 𝐜𝐨𝐬 Derivative 𝒙 𝟏 𝒇′ 𝒙 = 𝒙 3 𝒇 𝒙 = 𝐭𝐚𝐧 𝟏 𝒙 4 𝒇 𝒙 = 𝐜𝐨𝐭 𝟏 𝒙 5 𝒇 𝒙 = 𝐬𝐞𝐜 𝟏 𝒙 6 𝒇 𝒙 = 𝐜𝐨𝐬𝐞𝐜 𝟏 𝒙 𝒇′(𝒙) = − 𝟏 𝟏 − 𝒙𝟐 𝟏 𝟏 − 𝒙𝟐 𝟏 𝒇′ 𝒙 = 𝟏 + 𝒙𝟐 𝟏 𝒇′ 𝒙 = − 𝟏 + 𝒙𝟐 𝟏 𝒇′ 𝒙 = 𝒙 𝒙𝟐 − 𝟏 𝟏 𝒇′(𝒙) = − 𝟐 𝟏 Examples 1. 𝑦= Solution: 𝟏 + 𝟏 find derivative? Given, 𝑦 = 𝐬𝐢𝐧 𝟏 𝒙 + 𝒄𝒐𝐬 𝟏 𝒙 Taking derivative with respect to x 𝑑𝑦 𝑑 𝑑 𝟏 = 𝐬𝐢𝐧 𝒙 + 𝒄𝒐𝐬 𝟏 𝒙 𝑑𝑥 𝑑𝑥 𝑑𝑥 Using formula of derivative, such as = 2) 𝑦 = 𝒕𝒂𝒏 𝟏 𝒙 + 𝒄𝒐𝒕 𝟏 𝒙 find derivative? Solution: 𝟏 𝟏 𝒙𝟐 𝟏 − 𝟏 𝒙𝟐 (Ans) 𝟏 Given, 𝑦 = 𝟏 Taking derivative with respect to x 𝑑𝑦 𝑑 = 𝑑𝑥 𝑑𝑥 Using formula of derivative, such as = 𝟏 𝟏 𝟏 𝒙𝟐 − 𝟏 𝟏 𝟏 𝒙𝟐 (Ans) 3) Solution: 𝟏 𝟏 find derivative? 𝟏 Given, Taking derivative with respect to x 𝟏 𝟏 𝟏 Using formula of derivative, such as 𝒙 𝟏 𝟏 𝟏 4) Solution: Taking derivative with respect to x 𝟏 𝟏 𝒙𝟐 𝟏 𝟏 𝒙𝟐 𝟏 𝟏 𝒙𝟐 𝒙𝟐 𝟏 𝒙 𝟏 𝟏 + 𝟏 𝟏 + 𝟏 𝒙𝟐 𝟏 𝒙𝟐 𝟏 (Ans) + 𝟏 𝟏 𝟏 𝒙 𝒙𝟐 𝟏 𝒙 𝒙𝟐 𝟏 find derivative? (Ans) Home work 𝟏 𝟏 𝟏 𝟏 𝟏 𝟏 𝟏 𝟏 𝟏 derivative derivative derivative derivative derivative Simple function