The Chain Rule Section 3.4

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

Using the Chain Rule Find the derivative of each composition function. 𝑦= 𝑥 2 +1 3 𝑦= 𝑥 2 −1 2 3 𝑓 𝑥 = 3𝑥−7

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 2 5𝑥−7

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

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.