Math Problem Statement
Solution
We are tasked with finding the derivative using the alternative form of the derivative. The function is given as:
where .
The alternative form of the derivative is defined as:
We need to apply this formula with .
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Step 1: Calculate
Since , we substitute into the function: -
Step 2: Set up the derivative limit
Now apply the formula for the derivative:
Substituting the expression for and :
Simplifying:
-
Step 3: Factor the numerator
We can factor to simplify the limit:
Thus, the expression becomes:
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Step 4: Cancel common factors
The terms cancel out:
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Step 5: Evaluate the limit
Now we substitute into the remaining expression:
Thus, the derivative at is:
Would you like more details on any of the steps? Here are a few related questions:
- How do you use the first principles to find the derivative in general?
- What is the geometric interpretation of the derivative?
- Can the derivative of all functions be found using the alternative form?
- How do you handle discontinuities when using limits to find derivatives?
- What are some common mistakes when applying the alternative form of the derivative?
Tip: Factoring is often crucial when simplifying limits involving polynomials. Always check for common factors in the numerator and denominator.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Limits
Formulas
f'(c) = lim(x -> c) [(f(x) - f(c)) / (x - c)]
f(x) = x^3 + 3x^2 + 6
Theorems
Alternative form of the derivative
Suitable Grade Level
Grades 11-12 (Calculus)
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