Math Problem Statement
Solution
The problem asks to find the derivative of the function at , using the alternative form of the derivative.
Let's break this down step-by-step.
Function:
Simplifying the terms:
The Alternative Form of the Derivative:
The alternative definition of a derivative at is given by: Here, .
Step 1: Calculate
Step 2: Set up the limit definition of the derivative
We need to evaluate the following limit: This becomes:
Step 3: Simplify the expression
Notice that is a difference of squares, so we can factor it:
Now, cancel out the terms:
Step 4: Evaluate the limit
Substitute :
Final Answer:
Let me know if you'd like further clarification or details on any step! Here are some follow-up questions to consider:
- How does the alternative form of the derivative differ from the basic limit definition?
- What is the significance of factoring in simplifying limits in derivative calculations?
- How can you handle cases where the derivative does not exist at a certain point?
- Can you apply this derivative method to other types of functions, like rational functions?
- What are the benefits of using the limit definition over other methods to find derivatives?
Tip: Always check if a function can be factored to simplify limits when using the definition of the derivative.
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Math Problem Analysis
Mathematical Concepts
Algebra
Calculus
Derivatives
Limits
Formulas
f'(c) = lim (x -> c) [f(x) - f(c)] / (x - c)
f(x) = 3x^2 + 6
Theorems
Limit Definition of the Derivative
Suitable Grade Level
Grades 11-12 (High School Calculus)
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