Answer:
General Formulas and Concepts:
<u>Pre-Algebra</u>
- Equality Properties
<u>Algebra I</u>
- Functions
- Function Notation
- Exponential Rule [Root Rewrite]:
<u>Algebra II</u>
- Logarithms and Natural Logs
- Logarithmic Property [Multiplying]:
- Logarithmic Property [Exponential]:
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative Property [Multiplied Constant]:
Derivative Property [Addition/Subtraction]:
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Chain Rule]:
Logarithmic Derivative:
Implicit Differentiation
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
<em /><em />
<em />
<u>Step 2: Rewrite</u>
- [Equality Property] ln both sides:
- Logarithmic Property [Multiplying]:
- Exponential Rule [Root Rewrite]:
- Logarithmic Property [Exponential]:
<u>Step 3: Differentiate</u>
- ln Derivative [Implicit Differentiation]:
- Rewrite [Derivative Property - Addition]:
- Rewrite [Derivative Property - Multiplied Constant]:
- ln Derivative [Chain Rule]:
- Rewrite [Derivative Property - Addition]:
- Basic Power Rule]:
- Simplify:
- Multiply:
- [Multiplication Property of Equality] Isolate <em>y'</em>:
- Substitute in <em>y</em>:
- [Brackets] Add:
- Multiply:
- Simplify [Exponential Rule - Root Rewrite]:
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Implicit Differentiation
Book: College Calculus 10e