# 6: Differentiation Rules and Applications


• 6.1: The Sine and Cosine Function
In this section, we are going to work to conjecture formulas for the sine and cosine functions, primarily through a graphical argument. To help set the stage for doing so, the following preview activity asks you to think about exponential functions and why it is reasonable to think that the derivative of an exponential function is a constant times the exponential function itself.
• 6.2: The Product and Quotient Rules
If a function is a sum, product, or quotient of simpler functions, then we can use the sum, product, or quotient rules to differentiate the overall function in terms of the simpler functions and their derivatives. The product and quotient rules now complement the constant multiple and sum rules and enable us to compute the derivative of any function that consists of sums, constant multiples, products, and quotients of basic functions we already know how to differentiate.
• 6.3: Derivatives of Other Trigonometric Functions
The derivatives of the other four trigonometric functions are derived. These four rules for the derivatives of the tangent, cotangent, secant, and cosecant can be used along with the rules for power functions, exponential functions, and the sine and cosine, as well as the sum, constant multiple, product, and quotient rules, to quickly differentiate a wide range of different functions.
• 6.4: The Chain Rule
In this section, we encountered the following important ideas: A composite function is one where the input variable x first passes through one function, and then the resulting output passes through another.
• 6.5: Derivatives of Functions Given Implicitely
Implicit Differentiation is used to identfy the derivative of a y(x) function from an equation where y cannot be solved for explicitly in terms of x, but where portions of the curve can be thought of as being generated by explicit functions of x. In this case, we say that y is an implicit function of x. The process of implicit differentiation, we take the equation that generates an implicitly given curve and differentiate both sides with respect to x while treating y as a function of x.

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