Math Problem Statement
On a piece of paper, determine from the first principles,
f′(x) if f(x)=−2x. Then based on your working, what is the correct expression for
f(x+h)−f(x)?
Solution
To determine the derivative of the function from first principles, we use the definition of the derivative:
Step 1: Find
Given , we substitute into the function:
Step 2: Calculate
Now, compute :
Simplify the expression:
The and cancel each other out:
So, the expression for is:
Step 3: Compute the Derivative Using the Limit Definition
Substitute the expression we found into the limit definition of the derivative:
Simplify the fraction:
Conclusion
The derivative of is .
Based on our working, the correct expression for is:
Would you like more details on any step?
Here are 5 related questions you might consider:
- How do you apply the first principle to find the derivative of more complex functions?
- Why does the expression for simplify to just a constant in this case?
- How can we interpret the derivative geometrically?
- What happens to the derivative if the function is ?
- How does this method of finding derivatives compare with using differentiation rules?
Tip: The derivative represents the rate of change of a function; understanding its geometric interpretation can enhance comprehension.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Limit Definition
Formulas
Definition of the derivative
Theorems
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Suitable Grade Level
Advanced High School