1. Inverse Trig Derivatives and Tangent Line Approximations

2. Derivatives of Inverse Trig
𝑑 𝑑𝑥 arcsin 𝑢 = 𝑢 ′ 1− 𝑢 2
𝑑 𝑑𝑥 [ arctan 𝑢]= 𝑢 ′ 1+ 𝑢 2
𝑑 𝑑𝑥 arcsec 𝑢 = 𝑢 ′ 𝑢 𝑢 2 −1
𝑑 𝑑𝑥 arccos 𝑢 = −𝑢 ′ 1− 𝑢 2
𝑑 𝑑𝑥 [ arccot 𝑢]= − 𝑢 ′ 1+ 𝑢 2
𝑑 𝑑𝑥 arccsc 𝑢 = − 𝑢 ′ 𝑢 𝑢 2 −1

3. Ex 2. Find the derivatives.
a) 𝑑 𝑑𝑥 [ arcsin 2𝑥 ]

4. Ex 2. Find the derivatives.
b) 𝑑 𝑑𝑥 arctan (3𝑥)

5. Ex 2. Find the derivatives.
c) 𝑑 𝑑𝑥 [ arcsin 𝑥 ]

6. Ex 2. Find the derivatives.
d) 𝑑 𝑑𝑥 [ arctan 𝑒 2𝑥 ]

7. Log Properties as Aids to Differentiate
Recall: ln 𝑥𝑦 = ln 𝑥 + ln 𝑦
2. ln 𝑥 𝑦 = ln 𝑥− ln 𝑦
3. ln 𝑥 𝑟 =𝑟 ln 𝑥

8. Ex 3. Find the derivatives.
a) 𝑓 𝑥 = ln 𝑥+1

9. Ex 3. Find the derivatives.
b) 𝑓 𝑥 = ln 𝑥 𝑥 𝑥 3 −1

10. Tangent Line Approximations
Ex: Given 𝑓 𝑥 =2 𝑥 2 −3𝑥+1, write an equation to the line tangent to 𝑓 at 𝑥=1 and use it to approximate 𝑓(1.1).

11. Tangent Line Approximations
Ex: Given 𝑓 2 =1 and 𝑓 ′ 2 =−3, approximate 𝑓 2.1