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 from first principles, we use the definition of the derivative:
Step 1: Compute
Given , we substitute into the function:
Step 2: Compute
Now, subtract from :
Simplifying this:
Step 3: Derive the expression for
Substitute the expression into the limit definition:
Simplifying:
So, the derivative of is .
Answer to the Question:
Based on the working, the correct expression for is:
Would you like to go over any details or have further questions?
Here are 5 related questions you might find useful:
- What is the geometric interpretation of the derivative ?
- How does the result change if ?
- Can you find the derivative of from first principles?
- How would the process differ if the function involved a higher degree polynomial?
- What are the implications of a constant derivative value for ?
Tip: When applying the first principles of calculus, always simplify the expression as much as possible before taking the limit.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Limits
Formulas
Definition of derivative
Theorems
-
Suitable Grade Level
Grades 11-12