What is the formula for the derivative of an inverse function?
$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$
If $g(x)$ is the inverse of $f(x)$, what is $g'(x)$?
$g'(x) = \frac{1}{f'(g(x))}$
What is the point-slope form equation of a line?
$y - y_1 = m(x - x_1)$
How do you find the inverse of a function?
Switch $x$ and $y$ in the equation and solve for $y$.
What is the derivative of $f(x) = 2x + 1$?
$f'(x) = 2$
What is the general form of a linear function?
$f(x) = mx + b$
How to find $f^{-1}(x)$ if $f(x) = ax + b$?
$f^{-1}(x) = \frac{x-b}{a}$
What is the chain rule?
$\frac{d}{dx} [f(g(x))] = f'(g(x)) * g'(x)$
What is the power rule?
$\frac{d}{dx} x^n = nx^{n-1}$
What is the derivative of a constant?
$\frac{d}{dx} c = 0$
How do you find $(f^{-1})'(a)$ given $f(x)$?
1. Find $f^{-1}(a) = b$. 2. Find $f'(x)$. 3. Evaluate $f'(b)$. 4. Calculate $(f^{-1})'(a) = \frac{1}{f'(b)}$.
How do you find the tangent line to $g(x)$ at $x=a$, where $g(x) = f^{-1}(x)$?
1. Find $g(a)$. 2. Find $g'(a) = \frac{1}{f'(g(a))}$. 3. Use point-slope form: $y - g(a) = g'(a)(x - a)$.
How to find $g'(a)$ using a table of values?
1. Find $x$ such that $f(x) = a$, so $g(a) = x$. 2. Find $f'(x)$ from the table. 3. Calculate $g'(a) = \frac{1}{f'(x)}$.
Given $f(x)$ and a point $(a, b)$ on $f^{-1}(x)$, how do you find the equation of the tangent line to $f^{-1}(x)$ at $(a, b)$?
1. Verify that $f(b) = a$. 2. Find $f'(x)$. 3. Evaluate $f'(b)$. 4. The slope of the tangent line is $\frac{1}{f'(b)}$. 5. Use point-slope form: $y - b = \frac{1}{f'(b)}(x - a)$.
How do you solve for $g'(x)$ if $g(x)$ is the inverse of $f(x)$ and $f(x)$ is a complex function?
1. Find $f'(x)$. 2. Express $g'(x)$ as $\frac{1}{f'(g(x))}$. 3. If needed, use implicit differentiation or other techniques to find $g(x)$ or simplify the expression.
How do you determine if an inverse function is differentiable?
Check if the derivative of the original function is non-zero at the corresponding point. If $f'(f^{-1}(a)) \neq 0$, then $f^{-1}(x)$ is differentiable at $x = a$.
How do you find the value of $(f^{-1})'(a)$ if you are only given a graph of $f(x)$?
1. Find the point on the graph of $f(x)$ where $y = a$. Let this point be $(b, a)$. 2. Estimate the slope of the tangent line to $f(x)$ at $x = b$. This is $f'(b)$. 3. Calculate $(f^{-1})'(a) = \frac{1}{f'(b)}$.
How do you handle a problem where you need to find the derivative of a composite function involving an inverse function?
1. Apply the chain rule carefully, remembering that the derivative of the outer function is evaluated at the inner function. 2. Use the inverse derivative rule when differentiating the inverse function. 3. Simplify the expression.
How do you find the second derivative of an inverse function?
1. Find the first derivative $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$. 2. Differentiate this expression using the chain rule and quotient rule. 3. Simplify the result.
How do you find the derivative of an inverse trigonometric function?
Use the formula for the derivative of an inverse function and the derivatives of trigonometric functions. For example, $(\sin^{-1}(x))' = \frac{1}{\sqrt{1 - x^2}}$.
Explain the relationship between the derivatives of a function and its inverse.
The derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. If $f(a) = b$, then $(f^{-1})'(b) = \frac{1}{f'(a)}$.
How are the graphs of a function and its inverse related?
The graphs of a function and its inverse are reflections of each other across the line $y = x$.
What does the derivative of a function represent graphically?
The derivative of a function at a point represents the slope of the tangent line to the function's graph at that point.
Why is it important to know if a function is strictly increasing or decreasing when finding its inverse?
A strictly increasing or decreasing function is guaranteed to be one-to-one, and therefore invertible.
What is the significance of $f'(f^{-1}(x))$ in the inverse function derivative formula?
It represents the derivative of the original function evaluated at the inverse function, ensuring the correct corresponding point is used for the reciprocal calculation.
Explain the concept of local linearity.
At a sufficiently small scale, a differentiable function can be approximated by its tangent line.
What is the relationship between a function's domain and its inverse's range?
The domain of $f(x)$ is the range of $f^{-1}(x)$, and the range of $f(x)$ is the domain of $f^{-1}(x)$.
Explain the importance of differentiability when finding the derivative of an inverse function.
The original function must be differentiable at the point corresponding to the inverse function's input for the inverse derivative to exist.
What is the difference between $f(x)$ and $f^{-1}(x)$?
$f(x)$ is the original function, and $f^{-1}(x)$ is its inverse, which 'undoes' the operation of $f(x)$.
What is the difference between $f'(x)$ and $(f^{-1})'(x)$?
$f'(x)$ is the derivative of the original function, and $(f^{-1})'(x)$ is the derivative of its inverse.
