Finding The Derivative Of An Inverse Function: A Step-by-Step Guide

Let's dive into a classic calculus problem involving inverse functions and their derivatives. This is a common topic in AP Calculus, so understanding the concepts thoroughly is super important, guys. We're given a differentiable function $f$ with some specific values and derivatives, and we're asked to find the derivative of its inverse function, $g$, at a particular point. Let's break it down step by step so you can nail these types of questions.

Understanding the Problem

At its heart, this question tests your understanding of inverse functions and the relationship between the derivative of a function and the derivative of its inverse. Remember that if $g(x)$ is the inverse of $f(x)$, then $f(g(x)) = x$. This little nugget is the key to unlocking the solution, so keep it locked in your memory bank. The main challenge here is figuring out how to use the given information – the values of $f$ and $f'$ at specific points – to find $g'(3)$. It's like a detective puzzle, and we're the sleuths!

Key Concepts to Remember

Before we jump into the solution, let's quickly recap some crucial concepts:

• Inverse Functions: If $f(a) = b$, then $f^{-1}(b) = a$. This means the inverse function "undoes" what the original function does.
• Derivative of an Inverse Function: If $g(x) = f^{-1}(x)$, then $g'(x) = \frac{1}{f'(g(x))}$. This formula is the golden ticket to solving this problem, so make sure you're comfortable with it.
• Chain Rule: Remember the chain rule! It's essential for differentiating composite functions. In this case, it helps us derive the formula for the derivative of an inverse function.

With these concepts in mind, we're ready to tackle the problem head-on.

Solving for g'(3)

The million-dollar question is: how do we find $g'(3)$? We'll use the formula for the derivative of an inverse function we just discussed. Here’s the breakdown:

1. Apply the Formula: We know that $g'(x) = \frac{1}{f'(g(x))}$. So, to find $g'(3)$, we need to find $\frac{1}{f'(g(3))}$.
2. Find g(3): This is the crucial step. We need to determine what $g(3)$ is. Remember that $g(x) = f^{-1}(x)$. So, $g(3) = f^{-1}(3)$. This means we're looking for the value of $x$ such that $f(x) = 3$. Looking at the given information, we see that $f(6) = 3$. Therefore, $g(3) = f^{-1}(3) = 6$. See how we used the inverse function property to our advantage?
3. Find f'(g(3)): Now that we know $g(3) = 6$, we need to find $f'(g(3))$, which is $f'(6)$. The problem statement gives us that $f'(6) = -2$.
4. Calculate g'(3): Finally, we can plug the values into our formula: $g'(3) = \frac{1}{f'(g(3))} = \frac{1}{f'(6)} = \frac{1}{-2} = -\frac{1}{2}$.

And there you have it! We've successfully found that $g'(3) = -\frac{1}{2}$.

Step-by-Step Breakdown with Examples

Let's go through another example to really solidify this concept. Imagine we have a function $h(x)$ and its inverse $k(x) = h^{-1}(x)$. Suppose $h(2) = 5$ and $h'(2) = 4$. Let's find $k'(5)$.

1. Apply the Formula: $k'(x) = \frac{1}{h'(k(x))}$, so $k'(5) = \frac{1}{h'(k(5))}$.
2. Find k(5): Since $k(5) = h^{-1}(5)$, we need to find the $x$ value such that $h(x) = 5$. We know $h(2) = 5$, so $k(5) = 2$.
3. Find h'(k(5)): We need $h'(2)$, which is given as 4.
4. Calculate k'(5): $k'(5) = \frac{1}{h'(2)} = \frac{1}{4}$.

See the pattern? By carefully applying the formula and using the properties of inverse functions, you can solve these problems like a pro.

Common Mistakes to Avoid

Now, let's talk about some pitfalls that students often stumble into when dealing with these types of problems. Avoiding these mistakes can save you major headaches on the exam, trust me!

• Forgetting the Formula: The most common mistake is simply forgetting the formula for the derivative of an inverse function. Make sure you memorize it: $g'(x) = \frac{1}{f'(g(x))}$. Write it down, say it out loud, whatever it takes to make it stick!
• Confusing g(x) with 1/f(x): Remember, $g(x) = f^{-1}(x)$, which is the inverse function, not the reciprocal. Don't fall into the trap of thinking $f^{-1}(x) = \frac{1}{f(x)}$.
• Incorrectly Evaluating g(3): This is where careful reading and understanding of inverse functions come in. Remember that $g(3) = f^{-1}(3)$ means you're looking for the $x$ value that makes $f(x) = 3$. Don't mix up the inputs and outputs!
• Plugging in the Wrong Value into f'(x): Make sure you're evaluating $f'$ at the correct point, which is $g(x)$, not just $x$. This is a common slip-up, so double-check your work.

By being aware of these common errors, you can consciously avoid them and boost your accuracy.

Practice Problems

Okay, guys, now it's time to put your newfound knowledge to the test. Practice makes perfect, and the more you work through these problems, the more comfortable you'll become. Here are a couple of practice problems to get you started:

Problem 1:

Let $f(x)$ be a differentiable function with $f(4) = 2$ and $f'(4) = 5$. If $g(x)$ is the inverse of $f(x)$, find $g'(2)$.

Problem 2:

Given $f(x) = x^3 + 2x - 1$, find the derivative of its inverse function at $x = 2$. (Hint: You'll need to find the $x$ value such that $f(x) = 2$ first).

Work through these problems carefully, and don't be afraid to refer back to the steps we discussed earlier. The key is to break down the problem, identify the relevant information, and apply the formula correctly.

Conclusion

Understanding the derivative of an inverse function is a critical skill in calculus. By mastering the formula, avoiding common mistakes, and practicing regularly, you'll be well-equipped to tackle these types of problems on exams and beyond. Remember the key concepts, stay sharp, and keep practicing. You've got this! Now go out there and conquer those calculus challenges!