Rizzi – Calc BC
Previously On…

 Concept of and relationship between y=sinx and
y=arcsinx
 Graph of y=sinx and y=arcsinx
Proof of Derivative

 Let’s take the derivative of 𝑦 = sin−1 𝑥
 Start by letting 𝑥 = 𝑠𝑖𝑛𝑦
Now You Try

 Try taking the derivative of 𝑦 = tan−1 𝑥
But what happens
when…

 What if the inside isn’t just x?
 Let’s try it with 𝑦 = tan−1 (2𝑥 − 1)
 Can we come up with a formalized rule for the
derivative in words and in a formula?
Now You Try

 Try it with 𝑦 = 3 sin−1 (𝑥 2 )
 Write the rule in words and in a formula
Practice Problems

Do the following practice problems
from the book:
5.6 #39, 43, 59, 61
Rizzi – Calc BC
Let’s Work Backwards

Using our rules for derivatives and
knowledge of u-substitutions, let’s try to
work backwards to integrate the following.
First, identify whether it’s arcsin or arctan
𝑒𝑥
1 − 𝑒 2𝑥
𝑑𝑥
Together…

 What about this one?
12
𝑑𝑥
1 + 9𝑥 2
Try These…

𝑑𝑥
1+ 𝑥+3
𝑒 −𝑥
2
1 − 𝑒 −2𝑥
𝑑𝑥
And One More…

This one requires some fancy
simplifying first:
𝑑𝑥
9 − 𝑥2
Practice Problems

Do the following practice problems
from the book:
5.7 #1, 7, 9, 27, 43