Problem 264 - Calculus Volume 1

Question

Use the functions \(y=f(x)\) to find a. \(\frac{d f}{d x}\) at \(x=a\) and b. \(\quad x=f^{-1}(y)\) c. Then use part b. to find \(\frac{d f^{-1}}{d y}\) at \(y=f(a)\). $$ f(x)=6 x-1, x=-2 $$

Step-by-Step Solution

Answer

a. 6; b. \(f^{-1}(y) = \frac{y+1}{6}\); c. \(\frac{1}{6}\).

Step 1: Differentiate f(x)

The function given is \( f(x) = 6x - 1 \). To find \( \frac{d f}{d x} \), take the derivative of \( f(x) \). The derivative of \( 6x \) is 6, and since \(-1\) is a constant, its derivative is 0. Thus, \( \frac{d f}{d x} = 6 \).

Step 2: Evaluate the Derivative at x = a

Since \( a = -2 \), substitute \( x = -2 \) into the derivative to find \( \frac{d f}{d x} \) at \( x = -2 \). Therefore, \( \frac{d f}{d x} = 6 \) regardless of the value of \( x \).

Step 3: Find the Inverse Function f^{-1}(y)

To find the inverse function, switch \( x \) and \( y \) in the equation \( y = 6x - 1 \) and solve for \( x \). The equation becomes \( x = \frac{y + 1}{6} \). Therefore, \( f^{-1}(y) = \frac{y + 1}{6} \).

Step 4: Differentiate the Inverse Function f^{-1}(y)

Now, differentiate \( f^{-1}(y) = \frac{y + 1}{6} \) with respect to \( y \). The derivative of \( \frac{y}{6} + \frac{1}{6} \) is \( \frac{1}{6} \) because the derivative of \( y \) is 1 and the constant term drops out.

Step 5: Evaluate the Derivative at y = f(a)

Calculate \( y = f(a) \) using \( a = -2 \). The function \( f(x) = 6x - 1 \) when \( x = -2 \) is \( f(-2) = 6(-2) - 1 = -12 - 1 = -13 \). Evaluate \( \frac{d f^{-1}}{d y} \) at \( y = -13 \). Since the derivative of the inverse function is \( \frac{1}{6} \), it remains \( \frac{1}{6} \) for any \( y \).

Key Concepts

Derivative

A derivative is a way to show how a function changes at any point. It’s like looking at how steep a hill is at different places on your hike. The steeper the hill, the faster your position is changing. For mathematical functions, this change is described in terms of the slope of the tangent line at that point.For a linear function such as the one given, \( f(x) = 6x - 1 \), finding the derivative is especially simple. The process involves differentiation, which finds how much the function increases or decreases as \( x \) changes slightly.

• The derivative of a constant like \(-1\) is always zero, since a constant does not change as \( x \) changes.
• The derivative of \( 6x \) is just 6, since \( x \) changes linearly with a coefficient of 6.

Overall, the derivative \( \frac{df}{dx} \) gives us the slope of the function \( f(x) \) at any point \( x \). In this case, it remains the constant 6, meaning the function increases by 6 units for every one unit increase in \( x \).

Inverse Function Derivative

An inverse function essentially reverses what the original function does. If you think of a function as a machine that takes inputs (\( x \)), performs some operation, and gives an output (\( y \)), the inverse function takes the output and retrieves the original input.In our example, to find the inverse function \( f^{-1}(y) \), we swapped \( x \) and \( y \) in the equation \( y = 6x - 1 \) and solved for \( x \), giving \( f^{-1}(y) = \frac{y + 1}{6} \).When finding the derivative of an inverse function, we are looking to see how the inverse changes as \( y \) changes. Differentiating \( f^{-1}(y) = \frac{y + 1}{6} \) with respect to \( y \) gives you \( \frac{1}{6} \).

• This derivative is consistent for any \( y \), showing that the inverse function increases by \( \frac{1}{6} \) for every one unit increase in \( y \).
• The fact that the inverse’s derivative remains constant implies that retrieving the input from the output does not change based on where you are in the range of the function.

Function Differentiation

Differentiation is the process of finding the derivative of a function. It tells us about the function’s rate of change at any specific point. This is a core concept in calculus because it provides insight into how a function behaves.Differentiation often involves rules applied to elementary functions, such as polynomials. For the function \( f(x) = 6x - 1 \), applying the power rule, we see:

• For constants like \( -1 \), the derivative is zero.
• For linear terms like \( 6x \), the derivative results in the coefficient, which here is 6.

Finding the derivative of both the original and inverse functions, as seen in our example, helps to understand both the direct behavior of the function and its reversing capability. Knowing how to perform these derivatives allows for the analysis of more complex functions, enhancing the understanding of their behavior in different situations.