Problem 43 - AP Calculus with 8 Practice Tests 15th

Question

If \(h\) is the inverse function of \(f\) and if \(f(x)=\frac{1}{x},\) then \(h^{\prime}(3)=\) (A) -9 (B) \(-\frac{1}{9}\) (C) \(\frac{1}{9}\) (D) 9

Step-by-Step Solution

Answer

The answer is \(h^{\prime}(3) = -\frac{1}{9}\), option (B).

Step 1: Identify the given function and its inverse

We are given \(f(x) = \frac{1}{x}\). To find the inverse of this function, we denote the inverse as \(h(x)\). For \(h(x)\) to be the inverse, it must satisfy \(f(h(x)) = x\) and \(h(f(x)) = x\). Therefore, we conclude \(h(x) = \frac{1}{x}\), as inverting \(f(x)\) forms the same function.

Step 2: Differentiate the inverse function

Since \(h(x) = \frac{1}{x}\), the derivative \(h^{\prime}(x)\) is \(-\frac{1}{x^2}\). This is computed using the power rule, treating \(x\) as \(x^{-1}\) which differentiates to \(-1\) times \(x^{-2}\) which simplifies to \(-\frac{1}{x^2}\).

Step 3: Calculate the derivative at the specific point

We must find \(h^{\prime}(3)\), which is \(-\frac{1}{3^2} = -\frac{1}{9}\). Plugging \(3\) into the derivative \(-\frac{1}{x^2}\), the result is \(-\frac{1}{9}\).

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function. A derivative represents the rate of change of a function with respect to a variable. In simpler terms, it tells us how a function's output changes as we slightly change its input value.
Imagine you're on a road trip, and you want to know how fast you're going. Your speedometer gives you this information by showing how your distance changes over time. This is very much like a derivative. Differentiation provides a way to calculate this in mathematical terms.
To differentiate a function like \( f(x) = \frac{1}{x} \), we first rewrite it in a power form: \( x^{-1} \). The power rule, a fundamental rule in differentiation, states that the derivative of \( x^n \) is \( nx^{n-1} \). Applying this, the derivative of \( f(x) \) becomes \(-1 \cdot x^{-2} = -\frac{1}{x^2}\). This process is essential for solving problems related to rates and changes across science and engineering fields.

Derivative of Inverse Function

When dealing with inverse functions, understanding their derivatives can be somewhat tricky. An inverse function, denoted usually as \( h(x) \), is one that reverses the effect of another function \( f(x) \). For example, if \( f(x) = \frac{1}{x} \), then \( h(x) \), the inverse, is also \( \frac{1}{x} \).
To find the derivative of an inverse function, we use the relationship between a function and its inverse. If \( h \) is the inverse of \( f \), then the derivative of \( h \) at a point \( x \), \( h'(x) \), is the reciprocal of the derivative of \( f \) at \( h(x) \). Mathematically, this is written as:

• \( h'(x) = \frac{1}{f'(h(x))} \)

So, if \( f(x) = \frac{1}{x} \), the derivative \( f'(x) = -\frac{1}{x^2} \) and thus, \( h'(x) = -\frac{1}{x^2} \) as well, because \( h(x) = f(x) \). This concept helps us swiftly determine derivatives of inverse functions.

Calculus Problem-Solving

Calculus problem-solving involves using calculus tools like differentiation to solve complex mathematical problems. The problem provided is a classic example of this, involving the differentiation of inverse functions. To solve it, we break the issue into clear steps.
First, identify the function and its inverse. Here, both are \( \frac{1}{x} \). Then, compute the derivative using differentiation rules like the power rule. You obtain \( -\frac{1}{x^2} \).
Next, apply this derivative to the specific value given in the problem: compute \( h'(3) \) where \( h'(x) = -\frac{1}{x^2} \). Plugging \( 3 \) into this expression gives us:

• \(h'(3) = -\frac{1}{3^2} = -\frac{1}{9}\)

Thus, the solution to the problem is option \( B \), \(-\frac{1}{9}\). These steps illustrate effective calculus problem-solving by systematically applying differentiation concepts to reach a solution.