Math Problem Statement
Solution
We are given a table for a differentiable and increasing function and its derivative . We are asked to find , where is the inverse function of .
Key Information:
- The derivative of the inverse function can be expressed as:
- We are asked to find , meaning we need to find .
Steps:
- Since , we know .
- To find , first locate the value of such that . From the table, , so , meaning .
- Now, use the formula for the derivative of the inverse:
- From the table, , so:
Thus, the correct answer is option C.
Would you like any further clarifications or details?
Here are 5 related questions to expand your understanding:
- How do you compute the derivative of an inverse function in general?
- What are the conditions for a function to have an inverse?
- How does the relationship between a function and its inverse affect their derivatives?
- Can this method of finding the derivative of an inverse be applied to non-differentiable functions?
- What would change in the calculation if the function were not increasing?
Tip: When working with inverse functions, always check how the function and its inverse are related through specific values, especially at critical points.
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Derivatives
Chain Rule
Formulas
g'(x) = 1 / f'(g(x))
Theorems
Derivative of an inverse function
Suitable Grade Level
Grades 11-12
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