Math Problem Statement
for f(x) = x^2 + 3x + 1 find f'(1) using f'(a)= lim x->a (f(x)-f(a))/(x-a)
Solution
To find using the definition of the derivative, which is:
In this case, , so we need to compute . The function is , and we proceed with the following steps:
Step 1: Compute
We first evaluate :
Step 2: Write the difference quotient
Now, we use the difference quotient formula:
Substitute and :
Step 3: Simplify the expression
Simplify the numerator:
Now, factor the quadratic :
So the expression becomes:
Step 4: Cancel common terms
Cancel the terms:
Step 5: Evaluate the limit
Now, evaluate the limit as :
Thus, .
Would you like further clarification, or details on any of the steps?
Here are five related questions to deepen your understanding:
- How would you compute for any general value of using this same process?
- How does the result compare with using basic derivative rules like the power rule for this problem?
- What happens when a quadratic function is used in the limit definition of the derivative?
- How would you approach a similar problem if the function were cubic instead of quadratic?
- Can you think of a case where factoring might not work directly, and how could you handle it?
Tip: Always try simplifying the numerator in a difference quotient by factoring to cancel common terms with the denominator.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Limits
Derivatives
Quadratic Functions
Formulas
f'(a) = lim (x → a) (f(x) - f(a)) / (x - a)
Theorems
Limit definition of the derivative
Suitable Grade Level
Grades 11-12
Related Recommendation
Find f'(2) for f(x) = x^2 + 5x + 1 Using Limit Definition
Compute f'(-1) Using the Definition of the Derivative for f(x) = 2x^2 + 2x + 3
Finding the Derivative of f(x) = x^3 + x^2 + 1 Using the Definition of Derivative
Find the Derivative of f(x) = 3x + 1 at a = 5 Using the Limit Definition
Finding the Derivative of f(x) = 3x − 2 at x = 1 Using the Definition of a Derivative