Question

Finding a Second Derivative In Exercises \(85-90\) , find the second derivative of the function. $$ f(x)=\frac{8}{(x-2)^{2}} $$

Short answer

The second derivative of the function \(f(x)=\frac{8}{(x-2)^2}\) is \(f''(x)=\frac{48}{(x-2)^4}\).

Step-by-step solution

Find the First Derivative

The first derivative of \(f(x)=\frac{8}{(x-2)^{2}}\) can be found using the chain rule and the power rule, because the function is a composition of \(u(x) = x - 2\) and \(v(u) = \frac{8}{u^2}\). We get \(f'(x)=-\frac{16}{(x-2)^3}\) after deriving.

Taking the Second Derivative

Take the derivative of the first derivative \(f'(x)=-\frac{16}{(x-2)^3}\) to get the second derivative. Again, this will involve the chain rule and power rule. The resulting second derivative is \(f''(x)=\frac{48}{(x-2)^4}\).

Simplify the Solution

The result from the previous step is already fully simplified, so our final answer should be in this form.

Key concepts

Chain Rule

Understanding the chain rule is essential when dealing with complex functions that are composed of multiple functions within each other. The chain rule states that to find the derivative of a composed function, you multiply the derivative of the outer function by the derivative of the inner function.

Consider the function given in the exercise, \( f(x) = \frac{8}{(x - 2)^{2}} \). This function is essentially a composition of an inner function \( u(x) = x - 2 \) and an outer function \( v(u) = \frac{8}{u^2} \). Using the chain rule, the derivative of the outer function with respect to \( u \) is multiplied by the derivative of \( u(x) \). The process involves differentiating \( v(u) \) as \( v'(u) \) and \( u(x) \) as \( u'(x) \) then applying the chain rule: \( f'(x) = v'(u) \cdot u'(x) \).

Power Rule

The power rule is a basic yet powerful tool for computing the derivative of functions with exponents. It states that if you have a function in the form \( x^n \), where \( n \) is any real number, the derivative is \( nx^{n-1} \).

In our exercise, \( f'(x) = -\frac{16}{(x-2)^3} \) is derived by applying the power rule to the inner function \( (x-2)^{2} \) first, which gives us \( 2 \times \( (x-2)^{2-1} \) = 2(x-2) \) and then applying the negative because we're dealing with the reciprocal of \( (x-2)^2 \), resulting in \( -16(x-2)^{-3} \) before simplifying.

First Derivative Calculation

Calculating the first derivative is the process of determining the rate at which the function's output value changes with respect to changes in the input value. For our function \( f(x) = \frac{8}{(x - 2)^{2}} \), the first derivative \( f'(x) \) is found using the procedures outlined, involving both chain and power rules.

After applying these rules as explained in the previous sections, we obtain \( f'(x) = -\frac{16}{(x-2)^{3}} \). This represents the instantaneous rate of change of \( f(x) \) with respect to \( x \) and is crucial for understanding the behavior of the function at any given point.

Derivative Simplification

Simplifying the derivative involves rewriting the derivative in the simplest form possible, ensuring that it is easy to read and understand. Simplification might involve reducing fractions, factoring, and canceling terms.

In the context of our exercise, we find the simplified second derivative \( f''(x) \) directly after applying the differentiation rules. The resulting expression \( f''(x) = \frac{48}{(x-2)^4} \) is already in its simplest form. It’s important to note that simplification can be beneficial when evaluating derivatives at specific points or when integrating to find antiderivatives.