Math Problem Statement
Solution
The problem asks to find the derivative of at , using the table provided. We can use the quotient rule for derivatives to solve this.
Quotient Rule Formula:
For two functions and , the derivative of their quotient is:
In this case, and .
Step-by-Step Solution:
-
Identify the functions:
-
Find the derivatives:
-
Apply the Quotient Rule:
- Substitute :
From the table:
Substitute these values into the equation:
- Simplify:
- The denominator is
Thus, the derivative at is:
Final Answer:
Would you like a further breakdown of any steps?
Here are 5 related questions to deepen understanding:
- How does the quotient rule differ from the product rule in calculus?
- Can you explain why we need to square the denominator in the quotient rule?
- What would happen if the denominator was a more complex function?
- How do we handle higher-order derivatives using the quotient rule?
- In what scenarios is the quotient rule most useful in calculus problems?
Tip: Always verify that both the numerator and denominator functions are differentiable before applying the quotient rule.
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Math Problem Analysis
Mathematical Concepts
Calculus
Quotient Rule
Derivatives
Formulas
Quotient Rule: d/dx [u(x)/v(x)] = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2
Theorems
Quotient Rule
Suitable Grade Level
College level (Calculus I)
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