Problem 9 - Calculus for Biology and Medicine

Question

Use (4.12) to find the derivative of the inverse at the indicated point. Let $$f(x)=\sqrt{x+1}, \quad x \geq 0$$ Find \(\left.\frac{d}{d x} f^{-1}(x)\right|_{x=2} .[\) Note that \(f(3)=2 .\) ]

Step-by-Step Solution

Answer

The derivative of the inverse at \( x = 2 \) is 4.

Step 1: Understand the given function and point

We have the function \( f(x) = \sqrt{x+1} \) and are tasked with finding the derivative of its inverse at \( x = 2 \). We note that \( f(3) = 2 \), which implies that \( f^{-1}(2) = 3 \) since the inverse function swaps \( x \) and \( y \).

Step 2: Use the formula for the derivative of an inverse function

Recall that if \( y = f(x) \), then \( \frac{d}{dx}f^{-1}(y) = \frac{1}{f'(f^{-1}(y))} \). In this case, we have \( y = 2 \) and need to find \( f'(x) \) where \( x = f^{-1}(2) = 3 \).

Step 3: Find the derivative of the original function

We differentiate \( f(x) = \sqrt{x+1} \) with respect to \( x \):\[ f'(x) = \frac{1}{2\sqrt{x+1}}.\]

Step 4: Evaluate the derivative at the point x = 3

Substitute \( x = 3 \) into the derivative:\[ f'(3) = \frac{1}{2\sqrt{3+1}} = \frac{1}{2\cdot2} = \frac{1}{4}.\]

Step 5: Compute the derivative of the inverse at x = 2

Use the formula for the derivative of the inverse with \( f^{-1}(2) = 3 \):\[ \frac{d}{dx}f^{-1}(2) = \frac{1}{f'(3)} = \frac{1}{\frac{1}{4}} = 4.\]

Key Concepts

Derivative of Inverse Function

When dealing with inverse functions, it's important to understand how they relate to the concept of swapping inputs and outputs between two functions. An inverse function essentially reverses the operation of the original function. If you have a function \( f(x) \), its inverse is denoted by \( f^{-1}(x) \). When you input a value into an inverse function, it gives you back the original input for the corresponding output in \( f(x) \).The derivative of an inverse function at a certain point can be found using a specific formula:

• First, identify the point at which you need the inverse function's derivative, say \( y \).
• Next, calculate the derivative of the original function \( f'(x) \) at the point \( x = f^{-1}(y) \).
• Use the formula \( \left( \frac{d}{dx}f^{-1}(y) \right) = \frac{1}{f'(f^{-1}(y))} \) to find the derivative of the inverse function.

For the exercise provided, you found that \( f^{-1}(2) = 3 \) and \( f'(3) = \frac{1}{4} \). Thus, the derivative of the inverse at \( x = 2 \) is \( 4 \), as calculated by \( \frac{1}{\frac{1}{4}} = 4 \). This formula is crucial for practical use, especially in problems involving inverses.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents how a function's output value changes as the input changes. It's fundamentally about calculating the rate of change or slope of a curve at any given point.The differentiation process includes the following key aspects:

• Understanding the nature and structure of the function. For example, in the exercise with \( f(x) = \sqrt{x+1} \), we recognize it as a composition involving the square root.
• Applying differentiation rules to find the derivative. Common rules include power rules, product rules, and quotient rules.
• Using the extracted derivative to analyze the function further. In our case, \( f'(x) \) was found as \( \frac{1}{2\sqrt{x+1}} \), which shows how the square root function behaves in terms of rate of change.

Once the derivative is calculated, you can use it to make predictions and solve various mathematical problems such as finding maximums, minimums, and points of inflection.

Chain Rule

The chain rule is a fundamental technique in calculus used for differentiating compositions of functions. If a function \( y = g(f(x)) \), where \( g \) is a function of \( f(x) \), the chain rule helps us find the derivative of the entire expression.The chain rule formula is:\[\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\]This formula allows us to effectively handle complex functions by breaking them down into simpler parts.Here’s how the chain rule works in practice:

• Identify the inner function \( f(x) \) and the outer function \( g(u) \), where \( u = f(x) \).
• Differentiating the outer function \( g(u) \) with respect to its argument \( u \), yielding \( g'(u) \).
• Differentiating the inner function \( f(x) \) with respect to \( x \), giving us \( f'(x) \).
• Multiply these derivatives as shown in the chain rule formula to get the overall derivative.

In the original problem, we used simpler differentiation instead of the chain rule because \( f(x) = \sqrt{x+1} \) was straightforward to differentiate by applying the power rule directly. However, understanding the chain rule is crucial for more complex expressions involving nested or composite functions.