Math Problem Statement
Solution
The function given is:
Part a: Finding the slope of the line tangent to the inverse function at the point .
The slope of the tangent line to the inverse function at a point can be found using the formula for the derivative of the inverse function:
For , since (meaning ), we need to find .
First, differentiate using the quotient rule:
Now, evaluate :
The slope of the inverse function's tangent line at is:
So, the slope .
Part b: Finding the equation of the line tangent to the inverse function at .
The equation of a line is given by the point-slope form:
Here, the slope and the point is . Substituting these values:
Simplifying:
Thus, the equation of the tangent line is:
Would you like more details or have any further questions?
Related questions:
- What is the geometric meaning of the slope of the inverse function's tangent line?
- How do you apply the quotient rule to find the derivative of rational functions?
- Why does the derivative of the inverse function depend on the derivative of the original function?
- What is the significance of the point in the context of the function and its inverse?
- How can you check if a function is invertible at a given point?
Tip: The relationship between a function and its inverse can often be understood by switching and and solving for one variable in terms of the other.
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Derivatives
Tangent Lines
Formulas
f'(x) = (5 + x^2)(0) - (6)(2x) / (5 + x^2)^2
f^-1(a)' = 1 / f'(f^-1(a))
Point-slope form: y - y_1 = m(x - x_1)
Theorems
Derivative of Inverse Function Theorem
Quotient Rule for Differentiation
Suitable Grade Level
Undergraduate Level (Calculus)