The document discusses rules for taking derivatives of trigonometric and exponential functions. It provides the derivatives of sine, cosine, and exponential functions, as well as examples of using these rules to find the derivatives of more complex functions that include trigonometric and exponential terms. Specifically, it states that the derivative of sine is cosine, the derivative of cosine is negative sine, and the derivative of an exponential function is itself. It then shows examples of finding the derivatives of f(x) = sin2x + cos 3x and f(x) = sin 2x/ex.

1. The succeeding differentiation rules involve the trigonometric functions and exponential
functions. Using the definition of derivative, the following rules can be derived.
DERIVATIVE OF TRIGONOMETRIC & EXPONENTIAL FUNCTION
DERIVATIVE OF TRIGONOMETRIC &
EXPONENTIAL FUNCTION
EXAMPLE
A. Derivative of Sine Function
If 𝑓(𝑥) = sin⁡(𝑥), then 𝑓′(𝑥) = cos⁡(𝑥)
𝑓(𝑥) = 2 sin 𝑥
𝑓′(𝑥) = 2(𝑐𝑜𝑠𝑥)
𝑓′(𝑥) = 2 cos 𝑥
B. Derivative of Cosine Function
If 𝑓(𝑥) = cos⁡(𝑥), then 𝑓′(𝑥) = −⁡sin⁡(𝑥)
𝑓(𝑥) = 2 cos 𝑥
𝑓′(𝑥) = 2(− sin 𝑥)
𝑓′
(𝑥) = −2 sin 𝑥
C. Derivative of Exponential Function
If 𝑓(𝑥) = 𝑒𝑥
, then 𝑓′(𝑥) = 𝑒𝑥
OTHER EXAMPLES IN DIFFERENTIAITON OF TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS
EXAMPLE #1: Solve the derivative of the function: 𝑓(𝑥) = sin2𝑥 + cos 3𝑥
EXAMPLE #2: Solve the derivative of the function: 𝑓(𝑥) =
sin 2𝑥
𝑒𝑥
𝒇(𝒙) = 𝐬𝐢𝐧 𝟐𝒙 + 𝐜𝐨𝐬 𝟑𝒙
𝒇′(𝒙) = 𝟐(𝐜𝐨𝐬 𝒙) + 𝟑(− 𝐬𝐢𝐧 𝒙)
𝒇′(𝒙) = 𝟐 𝐜𝐨𝐬 𝒙 − 𝟑 𝐬𝐢𝐧 𝒙
𝒇(𝒙) =
𝐬𝐢𝐧 𝟐𝒙
𝒆𝒙
𝒇′(𝒙) =
(𝒆𝒙)(𝟐𝒄𝒐𝒔⁡𝒙) − (𝒔𝒊𝒏⁡𝟐𝒙)(𝒆𝒙
)
(𝒆𝒙)𝟐
𝒇′(𝒙) =
𝟐𝒆𝒙
𝒄𝒐𝒔⁡𝒙 − 𝒆𝒙
𝒔𝒊𝒏⁡𝟐𝒙
𝒆𝟐𝒙