1. The Chain Rule
Section 3.4

2. The Chain Rule Find the derivative of 𝑦= sin (2𝑥) . Chain Rule
𝑑 𝑑𝑥 𝑓 𝑔 𝑥 = 𝑓 ′ 𝑔 𝑥 𝑔 ′ (𝑥)

3. Using the Chain Rule Find the derivative of each composition function.
𝑦= 𝑥
𝑦= 𝑥 2 −
𝑓 𝑥 = 3𝑥−7

4. Combining Rules Find the derivative of the following: 𝑓 𝑥 = 𝑥 3 𝑥 2 +4
𝑓 𝑥 = 𝑥 3 𝑥 2 +4
𝑦= cos 𝑥 𝑥−7

5. Repeated Chain Rule Calculate the first derivative of each function:
𝑓 𝑡 = 𝑠𝑖𝑛 3 4𝑡
𝑓 𝑥 =3 𝑠𝑒𝑐 2 𝜋𝑡−1

6. Application
Find an equation of the tangent line to the graph of 𝑓 𝑥 =2 sin 𝑥 + cos 2𝑥 at the point 𝜋,1 . Then determine all values of 𝑥 in the interval (0, 2𝜋) at which the graph of 𝑓 has a horizontal tangent line.