Problem 270 - Calculus Volume 1

Question

Find \(\left(f^{-1}\right)^{\prime}(a)\). $$ f(x)=x+\sqrt{x}, a=2 $$

Step-by-Step Solution

Answer

\((f^{-1})'(2)=\frac{2}{3}\).

Step 1: Express the Inverse Function

To find the derivative of the inverse function, first express the inverse function. The given function is \( f(x) = x + \sqrt{x} \). To find \( f^{-1}(x) \), let \( y = x + \sqrt{x} \) and solve for \( x \) in terms of \( y \).

Step 2: Differentiate the Original Function

First, differentiate \( f(x) = x + \sqrt{x} \). The derivative is \( f'(x) = 1 + \frac{1}{2\sqrt{x}} \).

Step 3: Evaluate the Derivative at the Expected Inverse Point

Since \( a = 2 \), we need to find \( x = f^{-1}(a) \), which implies \( f(x) = 2 \). Solve \( x + \sqrt{x} = 2 \), which gives \( x = 1 \). Therefore, evaluate \( f'(1) = 1 + \frac{1}{2\sqrt{1}} = 1.5 \).

Step 4: Use the Derivative of the Inverse Formula

The formula for the derivative of the inverse function at a point \( a \) is \( (f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))} \). Substituting \( f^{-1}(2) = 1 \) and \( f'(1) = 1.5 \), we have \( (f^{-1})'(2) = \frac{1}{1.5} = \frac{2}{3} \).

Key Concepts

Inverse Functions

Inverse functions are an essential concept in calculus, especially when dealing with function differentiation. An inverse function is essentially a function that "undoes" the action of the original function. If you have a function \( f(x) \), its inverse \( f^{-1}(x) \) satisfies the property that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
To find an inverse function, you typically solve for \( x \) in terms of \( y \) when the function is given in the form \( y = f(x) \). This often requires algebraic manipulation or solving equations to express \( x \) as a function of \( y \).
In practice, not all functions have inverses. A function must be bijective—both one-to-one and onto—to ensure that it has an inverse. This often involves checking the function's properties or imposing restrictions to ensure a unique solution exists for every output.

Derivative

The derivative of a function is a powerful tool in calculus used to understand how functions change. It provides the rate at which a function's value is changing at any given point.
Formally, the derivative of a function \( f(x) \), denoted as \( f'(x) \) or \( \frac{df}{dx} \), is defined as the limit:

• \( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \)

Calculating derivatives allows us to determine the slope of the tangent line to the graph of the function at any point \( x \). This can be used to identify features such as critical points, which may indicate local maxima, minima, or points of inflection.
Derivatives are fundamental to many calculus applications, from calculating velocity in physics to finding optimal solutions in economics.

Function Differentiation

Function differentiation refers to the process of computing the derivative of a function. It's about determining how a function's output value changes as its input changes.
In the context of the function \( f(x) = x + \sqrt{x} \), differentiating involves applying basic differentiation rules. These rules include:

• The power rule: \( \frac{d}{dx} x^n = nx^{n-1} \)
• The sum rule: \( \frac{d}{dx} [u(x) + v(x)] = u'(x) + v'(x) \)

Using these, the derivative of \( f(x) = x + \sqrt{x} \) is \( f'(x) = 1 + \frac{1}{2\sqrt{x}} \).
Differentiation is central in calculus because it enables us to analyze and predict the behavior of dynamic systems by understanding how changes in input affect the overall function.

Inverse Derivative Formula

The inverse derivative formula is a key concept when working with inverses of differentiable functions. When you know the derivative of a function \( f \), the inverse derivative formula allows you to find the derivative of its inverse.
This formula is particularly useful if directly differentiating the inverse function is complex or impractical. The formula is given by:

• \( (f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))} \)

In the exercise you're examining, you use this formula to find the derivative of the inverse function at a specific point \( a = 2 \). First, find \( f^{-1}(a) \), which is the value of \( x \) that makes \( f(x) = a \). Then, calculate \( f'(x) \) at this point and use the inverse derivative formula to get the answer.
This method simplifies finding the inverse derivative without explicitly finding the inverse function's form, which can often be quite challenging.