Math Problem Statement
Consider functions π(π₯) = β π₯ π π π=0 (= (π₯ π+1β1) π₯β1 , for π₯ β 1) and π(π₯) = (π₯ β 1) 2 . (a) Compute the first derivative πβ²(π₯) of function π(π₯) represented as a summation. (b) What is the value of π β² (π₯) for π₯ = 1 and π₯ β 1?
Solution
Given the functions:
and
(a) Compute the First Derivative of
Since , we can find using the quotient rule for derivatives. For and :
Using the quotient rule:
Expanding and simplifying:
(b) Compute for and
For :
Using the simplified expression obtained above, we have:
For :
To find , we use L'HΓ΄pital's rule, since the expression is indeterminate at .
- Differentiate the numerator and denominator:
- The derivative of is .
- The derivative of is .
Then:
So:
- For :
- For :
Would you like more details on this solution or have further questions?
Here are some follow-up questions you might consider:
- How is the quotient rule applied in general, and why is it useful here?
- What are the conditions for using L'HΓ΄pital's rule in calculating limits?
- How does the function relate to the behavior of as approaches 1?
- Can the derivative be interpreted graphically around ?
- What happens to as for various values of ?
Tip: When dealing with indeterminate forms, L'HΓ΄pital's rule is often a reliable tool to simplify limits.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Summation
L'HΓ΄pital's Rule
Quotient Rule
Formulas
f(x) = (x^(n+1) - 1) / (x - 1)
g(x) = (x - 1)^2
Quotient Rule: (h(x)/k(x))' = (h'(x)k(x) - h(x)k'(x)) / (k(x))^2
L'HΓ΄pital's Rule: lim(x β c) f(x)/g(x) = lim(x β c) f'(x)/g'(x) when lim(x β c) f(x)/g(x) is indeterminate
Theorems
Quotient Rule
L'HΓ΄pital's Rule
Suitable Grade Level
College Level - Calculus I
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