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Magazine Chapter 2: Problem 6
Find the second derivative. $$ f(x)=(3 x+2)^{-3} $$
Short Answer
Expert verified
The second derivative is \(f''(x) = 108(3x + 2)^{-5}\).
Step by step solution
01
Find the first derivative
Start by using the chain rule to find the first derivative of the function. The chain rule states that \(\frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x)\). In this case, let \(u = 3x + 2\), so the function becomes \(f(x) = u^{-3}\). The derivative of \(u^{-3}\) with respect to \(u\) is \(-3u^{-4}\), and the derivative of \(u\) with respect to \(x\) is \(3\). Therefore, \(\frac{d}{dx}[u^{-3}] = -3u^{-4} \cdot 3 = -9u^{-4}\). Substitute back \(u = 3x + 2\), we get \(f'(x) = -9(3x + 2)^{-4}\).
02
Simplify the first derivative
Simplify the expression \(f'(x) = -9(3x + 2)^{-4}\). The first derivative is simplified as: \(f'(x) = -9(3x + 2)^{-4}\).
03
Find the second derivative
Now, find the second derivative by differentiating \(f'(x) = -9(3x + 2)^{-4}\) with respect to \(x\). Again, apply the chain rule. Let \(v = 3x + 2\), so the function becomes \(f'(x) = -9v^{-4}\). The derivative of \(v^{-4}\) with respect to \(v\) is \(-4v^{-5}\), and the derivative of \(v\) with respect to \(x\) is still \(3\). Therefore, \(\frac{d}{dx}[v^{-4}] = -4v^{-5} \cdot 3 = -12v^{-5}\). Substitute back \(v = 3x + 2\), we get \(f''(x) = -12 \cdot -9(3x + 2)^{-5}\). Simplify to get \(\boxed{108(3x + 2)^{-5}}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Second Derivative
In calculus, the second derivative of a function is the derivative of the first derivative. It provides information on the concavity and the rate of change of the slope of the function. When you differentiate a function twice, you're essentially looking at how the rate at which your function is changing, is itself changing.
If you start with a function like \(f(x) = (3x + 2)^{-3}\), you first find the first derivative, and then you again differentiate the first derivative to get the second one.
Here's a quick walkthrough of our example:
If you start with a function like \(f(x) = (3x + 2)^{-3}\), you first find the first derivative, and then you again differentiate the first derivative to get the second one.
Here's a quick walkthrough of our example:
- The original function is \(f(x) = (3x + 2)^{-3}\).
- The first derivative, after using the chain rule, is \(f'(x) = -9(3x + 2)^{-4}\).
- To get the second derivative, we differentiate \(f'(x)\) again, using the chain rule, to get \(f''(x) = 108(3x + 2)^{-5}\).
- Helps in determining the concavity of the function.
- Useful in identifying points of inflection.
- Important for optimization problems to understand the nature of critical points.
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. If you have a function composed of another function (like in \(f(x) = (3x + 2)^{-3}\)), you use the chain rule to break it down.
In the given problem, we start by letting \(u = 3x + 2\). Here鈥檚 how the chain rule helps:
In the given problem, we start by letting \(u = 3x + 2\). Here鈥檚 how the chain rule helps:
- Differentiating \(u^{-3}\text{ with respect to }u\) gives us \(-3u^{-4}\).
- Then, we differentiate \(u\) with respect to \(x\), giving us \(3\).
- We multiply these results to find \(f'(x) = -9u^{-4}\).
- Finally, we substitute back \(u = 3x + 2\) to get \(f'(x) = -9(3x + 2)^{-4}\).
- Let \(v = 3x + 2\).
- Differentiating \(v^{-4}\) with respect to \(v\) results in \(-4v^{-5}\).
- Multiplying by the derivative of \(v\) with respect to \(x\), we get \(-12v^{-5}\).
- Substitute back \(v = 3x + 2\) and multiply the coefficients to get \(f''(x) = 108(3x + 2)^{-5}\).
Differentiation
Differentiation is the process of finding the derivative of a function, which shows the rate of change. In simple terms, if you have a function \(f(x)\), its derivative \(f'(x)\) tells you how \(f(x)\) changes as \(x\) changes.
To differentiate a function:
To differentiate a function:
- Identify the function and its components.
- Use differentiation rules like the power rule, product rule, quotient rule, or chain rule.
- Simplify the resulting expression.
- First, apply the chain rule to get the first derivative: \(f'(x) = -9(3x + 2)^{-4}\).
- Then differentiate again to get the second derivative: \(f''(x) = 108(3x + 2)^{-5}\).
- Helps in sketching curves and understanding their geometric properties.
- Essential for solving real-world problems involving rates of change in physics, biology, and economics.
- Used in optimization to find maximum and minimum values of functions.
