Question

Find the derivative of the inverse of the following finctions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\). $$f(x)=\frac{1}{2} x+8 ;(10,4)$$

Short answer

To summarize, the given function is \(f(x) = \frac{1}{2}x + 8\), and we are given a point on the graph of its inverse function, \((10,4)\). By differentiating the given function, we found that \(f'(x) = \frac{1}{2}\). Using the inverse function derivative formula, we found that the derivative of the inverse function at the specified point is \(2\).

Step-by-step solution

Find the derivative of the given function

To find the derivative \(f'(x)\) of the function \(f(x) = \frac{1}{2}x + 8\), we just differentiate each term with respect to \(x\). $$f'(x)=\frac{d}{dx}\left(\frac{1}{2}x+8\right)=\frac{1}{2}$$

Find the corresponding \(y\)-value

We are given a point on the graph of the inverse function, \((10,4)\). According to the property of inverse functions, this means that the graph of the function has a point \((4,10)\). Therefore, when we plug in \(x=4\) into the inverse function, we will get \(y=10\). In other words, \(f^{-1}(4)=10\).

Evaluate the derivative of the inverse function

Now we can evaluate the derivative of the inverse function using the formula: $$\left(f^{-1}(x)\right)'=\frac{1}{f'(f^{-1}(x))}$$ In our case, we have \(x=4\), and we know that \(f^{-1}(4)=10\). This means that we need to evaluate the derivative \(f'\) at the point where \(f(x)=10\) which is the value of the original function at \(x=4\). However, we already found that \(f'(x) = \frac{1}{2}\) for all \(x\). Therefore, we have: $$\left(f^{-1}(4)\right)'=\frac{1}{f'(10)}=\frac{1}{\frac{1}{2}}=2$$ The derivative of the inverse function at the specified point on the graph of the inverse function is 2.

Key concepts

Inverse Functions

Inverse functions are like mathematical mirrors. They flip the roles of inputs and outputs. This means if a function takes an input and gives an output, its inverse function will take this output and give back the original input. For example, if the function \(f(x) = \frac{1}{2}x + 8\) converts \(4\) to \(10\), the inverse function \(f^{-1}\) will convert \(10\) back to \(4\). For a pair of functions to be inverses, the original function applied to the inverse function must return the original value:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

In simpler terms, each function undoes what the other does. Understanding this relationship is crucial for solving problems related to inverse functions.

In our example, we know the inverse of \(f(x)\) because of the given point \((10,4)\) on the inverse function's graph. This gives us valuable information without actually calculating \(f^{-1}\).

Differentiation

Differentiation is the process used in calculus to find the derivative of a function. It measures how a function changes as its input changes. Think of the derivative as the function's rate of change or slope. For the function \(f(x) = \frac{1}{2}x + 8\), we can find its derivative by calculating \(f'(x)\). This derivative helps us understand how steep the function is at any point. For linear functions, like ours, this slope is constant. Hence, \(f'(x) = \frac{1}{2}\) at every point.

When dealing with inverse functions, knowing the original function's derivative is crucial. Why? Because it guides us in finding the derivative of the inverse, which tells us about the inverse's rate of change at a certain point. Use the rule:

• \(\left(f^{-1}(x)\right)' = \frac{1}{f'(f^{-1}(x))}\)

This formula shows that to find the derivative of the inverse, you need the derivative of the original function and the value from the inverse. It’s a neat mathematical trick to solve problems involving rates of change for inverse functions.

Mathematical Graphing

Mathematical graphing is a visual method of representing functions. It's immensely helpful in understanding what's happening with a function at different points. For instance, by plotting \(f(x) = \frac{1}{2}x + 8\), you can see that it's a straight line. Knowing this can help predict behavior, such as the constant slope indicated by the derivative.

Graphs of inverse functions are also insightful. The point \((10,4)\) on the graph of the inverse function tells us that \((4,10)\) lies on the graph of the original function. When graphing, inverse functions will often reflect over the line \(y = x\). This reflection showcases the swapping of \(x\) and \(y\). In practice, effective graphing skills can make solving complex calculus problems more intuitive and less intimidating.

Graphical representation of derivatives helps understand how changes in the input (\(x\)) affect changes in the function. These visual tools are a powerful asset in mathematics, aiding both in comprehension and problem-solving.