Unit 2: Differentiation-Limit Definition and Fundamental Properties

What is a Derivative?
What is a Derivative?
Algebra vs. Calculus
Is it Differentiable?
Derivative Notation and Language
Limit Definition of the Derivative (The Long Way)
4 Topics
Meaning: Limit Definition of the Derivative
Identifier: Limit Definition of the Derivative
Method: Limit Definition of the Derivative
Example 1: Limit Definition of the Derivative
The 5 Main Derivative Rules
5 Main Derivative Rules
Always Step 1: Simplify and Look for Algebraic Rewrites.
4 Topics
Roots or Radicals
x’s on the Bottom of a Fraction
Trig Functions Raised to a Power
Look to Make Your Life Easier
Rule 1: Constants
3 Topics
Identifier: Derivative of Constants
Method: Derivative of Constants
Example 1: Derivative of a Constant
Rule 2: Power Rule
5 Topics
Identifier: Power Rule Derivative
Method: Power Rule Derivative
Example 1: Power Rule Derivative
Example 2: Power Rule Derivative
Example 3: Power Rule Derivative
Rule 3: Product Rule
3 Topics
Identifier: Product Rule Derivative
Method: Product Rule Derivative
Example 1: Product Rule Derivative
Rule 4: Quotient Rule
3 Topics
Identifier: Quotient Rule Derivative
Method: Quotient Rule Derivative
Example 1: Quotient Rule Derivative
Rule 5: Chain Rule
3 Topics
Identifier: Chain Rule Derivative
Method: Chain Rule Derivative
Example 1: Chain Rule Derivative
Special Case Derivatives: Your new multiplication tables
Special Case Derivatives Overview
Six Trig Functions
4 Topics
Identifier: Trig Function Derivatives
Method: Trig Function Derivatives
Example 1: Trig Function Derivatives
Example 2: Trig Function Derivatives
Six Inverse Trig Functions
4 Topics
Identifier: Inverse Trig Derivatives
Method: Inverse Trig Derivatives
Example 1: Inverse Trig Derivatives
Example 2: Inverse Trig Derivatives
ex
3 Topics
Identifier: e^(x) Derivative
Method: e^(x) Derivatives
Example 1: e^(x) Derivative
ln ( x )
3 Topics
Identifier: ln(x) Derivative
Method: ln(x) Derivative
Example: ln(x) Derivative
a x = Any number raised to the x
3 Topics
Identifier: a^(x) Derivative
Method: a^(x) Derivative
Example 1: a^(x) Derivative
log b ⁡ x
3 Topics
Identifier: Logarithm Derivative
Method: Logarithm Derivative
Example 1: Logarithm Derivative
f – 1 (x) = Inverse Function
3 Topics
Identifier: Inverse Function Derivative
Method: Inverse Function Derivative
Example 1: Inverse Function Derivative
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Six Inverse Trig Functions

Unit 2: Differentiation-Limit Definition and Fundamental Properties Six Inverse Trig Functions

Base Case: Six Inverse Trig Functions

f ( x )

 

f ′ ( x )

f ( x ) = sin – 1 ( x )

f ( x ) = arcsin ( x )

 

f ′ ( x ) = 1 1 – x 2

f ( x ) = cos – 1 ( x )

f ( x ) = arccos ( x )

 

f ′ ( x ) = – 1 1 – x 2

f ( x ) = tan – 1 ( x )

f ( x ) = arctan ( x )

 

f ′ ( x ) = 1 1 + x 2

f ( x ) = sec – 1 ( x )

f ( x ) = arcsec ( x )

 

f ′ ( x ) = 1 | x | x 2 – 1

f ( x ) = csc – 1 ( x )

f ( x ) = arccsc ( x )

 

f ′ ( x ) = – 1 | x | x 2 – 1

f ( x ) = cot – 1 ( x )

f ( x ) = arccot ( x )

 

f ′ ( x ) = – 1 1 + x 2

 

Note :

–          Keep in mind that as soon as whatever is inside the function, inside the parenthesis, is more than just your basic x , the problem’s primary rule becomes a chain rule. Whatever you replace the x with inside your base special case will become the inside piece of your chain rule.

–          Also, when you apply the chain rule, everywhere you see an x in the f ′ ( x ) equations above, you will plug in the full inside piece.

–          You will need to know this table equally as well in both directions, left to right and right to left.

–          Conveniently math teachers have not come to an agreement on how to refer to inverse trig functions. So, depending on where you are taking your AP Calculus class you could see either version, (i.e.,  sin – 1 ( x )   or  arc sin ( x ) ) used to describe an inverse trig function. They both mean the same thing, and they both have the same derivative.

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