Question

Finding a Second Derivative In Exercises \(85-90\) , find the second derivative of the function. $$ f(x)=\frac{1}{x-6} $$

Short answer

The second derivative of the function \( f(x) = \frac{1}{x-6} \) is \( f''(x) = \frac{2}{{(x-6)}^3} \).

Step-by-step solution

Find the First Derivative

To find the first derivative, we rewrite the function in a form that makes it easier to differentiate. The function \( f(x) = \frac{1}{x-6} \) can be written as \( f(x) = (x-6)^{-1} \). Applying the power rule of differentiation (which states that the derivative of \( x^n \) is \( nx^{n-1} \)), we find that the first derivative, \( f'(x) \), is \( -1*(x-6)^{-2} \) or \( -\frac{1}{{(x-6)}^2} \).

Find the Second Derivative

The next move is to differentiate \( f'(x) = -\frac{1}{{(x-6)}^2} \) to find the second derivative. Again, we write this function in a form that is easier to differentiate, namely \( f'(x) = -(x-6)^{-2} \). By applying the power rule, and remembering to multiply by the derivative of the inside function because of the chain rule, we obtain that the second derivative, \( f''(x) \), is: \( 2*(x-6)^{-3} \) or \( \frac{2}{{(x-6)}^3} \).

Simplify the Result

The second derivative, \( f''(x) = \frac{2}{{(x-6)}^3} \), is the final answer. This function indicates how the slope of the original function changes for different values of x.

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that helps us understand how functions change.

• It allows us to find the rate at which a function is changing at any given point.
• By differentiating a function, we can obtain its derivative. This is represented by the symbol \( f'(x) \), which tells us the slope of the tangent line to the graph of the function at any point x.

For our exercise, we needed to find the second derivative of the function \( f(x) = \frac{1}{x-6} \).
To get there, we first had to find the first derivative and then differentiate it again. This two-step process of differentiation shows how effectively we can study changes in functions beyond the initial rate of change.

Power Rule

The power rule is an essential tool for differentiation. It makes finding derivatives of power functions straightforward.

• For any function of the form \( x^n \), where n is a constant, the derivative is \( nx^{n-1} \).

In our given function, \( f(x) = \frac{1}{x-6} \), rewriting it as a power function \( (x-6)^{-1} \) was necessary for simplification.
With the power rule, we quickly found the first derivative: \( f'(x) = - (x-6)^{-2} \).
This simplicity is what makes the power rule so powerful, allowing us to tackle differentiation problems effectively.

Chain Rule

The chain rule is a vital technique when dealing with complex functions, particularly in differentiation. It enables us to differentiate composite functions efficiently.

• In essence, the chain rule states that if a function y is composed of two functions, say g(x) and h(x), then the derivative is found by multiplying the derivative of the outer function by the derivative of the inner function.
• This is especially useful in our exercise as we moved to the second derivative.

While finding the second derivative \( f''(x) \), after getting \( f'(x) = - (x-6)^{-2} \), the chain rule helped us factor in the derivative of the inner function \( (x-6) \), which is simply 1.
This step involved the power rule combined with the chain rule to achieve the correct second derivative.

First Derivative

The first derivative is invaluable for understanding the basic behavior and slope of functions.

• It's the first step in determining how a function's value changes as x changes.
• In calculus, finding the first derivative positions us to explore further behaviors of the function more deeply.

For our specific problem, obtaining the first derivative involved rewriting \( f(x) = \frac{1}{x-6} \) as \( (x-6)^{-1} \) before differentiation.
By applying the power rule, we found \( f'(x) = - (x-6)^{-2} \).
This first derivative set the stage for finding the second derivative, which gives further insight into how the function accelerates or decelerates around certain values of x.