Math Problem Statement
Find the derivative of f(x)=x²+2x using the first principle.
Solution
The derivative of a function using the first principle (or definition of the derivative) is given by:
Given , let's calculate the derivative step-by-step.
Step 1: Compute
Substitute into the function :
Expanding the terms:
Step 2: Set up the difference
Now subtract from :
Simplify the expression:
Step 3: Divide by
Next, divide the expression by :
Simplify:
Step 4: Take the limit as
Finally, take the limit of the expression as :
Since , we get:
Thus, the derivative of using the first principle is:
Would you like more details or have any questions?
Relative questions:
- How is the first principle of differentiation different from using standard rules?
- Can you apply the first principle to find the derivative of ?
- What is the geometric interpretation of the derivative?
- How would the first principle work for non-polynomial functions, like ?
- What are the limitations of using the first principle for complex functions?
Tip:
The first principle of differentiation is a useful tool for understanding the concept of limits and how they apply to the slope of a function at a point.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
First Principle of Differentiation
Formulas
f'(x) = lim(h -> 0) [(f(x+h) - f(x)) / h]
Theorems
Definition of Derivative using First Principles
Suitable Grade Level
Grade 11-12