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Magazine Chapter 4: Problem 22
Find the derivative. $$ 1 / \sqrt{\ln x} $$
Short Answer
Expert verified
The derivative is \( f'(x) = -\frac{1}{2x(\ln x)^{3/2}} \).
Step by step solution
01
Understanding the Problem
We need to find the derivative of the function \( f(x) = \frac{1}{\sqrt{\ln x}} \). This means we are looking for \( f'(x) \), the rate of change of \( f(x) \) with respect to \( x \).
02
Rewrite the Function
Start by rewriting the function for easier differentiation. Since the square root can be expressed as a power of \( 1/2 \), we have: \( f(x) = (\ln x)^{-1/2} \).
03
Apply the Chain Rule
The function \( f(x) = (\ln x)^{-1/2} \) is a composition of functions. Let \( u = \ln x \), then \( f(x) = u^{-1/2} \). We will apply the chain rule, which states \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).
04
Differentiate Outer Function
Differentiate \( u^{-1/2} \) with respect to \( u \): \( \frac{d}{du}[u^{-1/2}] = -\frac{1}{2}u^{-3/2} \). So, our outer derivative is \( -\frac{1}{2}(\ln x)^{-3/2} \).
05
Differentiate Inner Function
Differentiate the inner function \( \ln x \) with respect to \( x \): \( \frac{d}{dx}[\ln x] = \frac{1}{x} \).
06
Apply the Chain Rule
Combine the derivatives using the chain rule: \( f'(x) = -\frac{1}{2}(\ln x)^{-3/2} \cdot \frac{1}{x} = -\frac{1}{2x(\ln x)^{3/2}} \).
07
Simplify the Result
Therefore, the derivative of \( f(x) = \frac{1}{\sqrt{\ln x}} \) is \( f'(x) = -\frac{1}{2x(\ln x)^{3/2}} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Derivatives
A derivative represents the rate of change of a function with respect to a variable. Imagine driving a car and looking at the speedometer. The speedometer shows your speed at a specific moment, which is analogous to finding a derivative鈥攊t tells you how fast your position is changing.
For a given function like \( f(x) = \frac{1}{\sqrt{\ln x}} \), finding the derivative, \( f'(x) \), tells us how the output of this function changes as \( x \) changes slightly. The main goal is to understand the behavior of the function as \( x \) varies.
Key points to remember about derivatives:
For a given function like \( f(x) = \frac{1}{\sqrt{\ln x}} \), finding the derivative, \( f'(x) \), tells us how the output of this function changes as \( x \) changes slightly. The main goal is to understand the behavior of the function as \( x \) varies.
Key points to remember about derivatives:
- The derivative of a function at a point provides the slope of the tangent line to the graph of the function at that point.
- Derivatives can show us both the instantaneous rate of change and overall change trends of a function.
- You can think of the derivative function \( f'(x) \) as a formula that gives the slope of the function at any \( x \), within its domain.
The Chain Rule Explained
The chain rule is an essential tool in calculus for computing derivatives of composite functions. When a function is nested inside another function, we frequently encounter this useful rule.
Consider the function \( f(x) = \frac{1}{\sqrt{\ln x}} \). We can think of this function as having an "inner function" and an "outer function":
Consider the function \( f(x) = \frac{1}{\sqrt{\ln x}} \). We can think of this function as having an "inner function" and an "outer function":
- Inner function: \( g(x) = \ln x \)
- Outer function: \( h(u) = u^{-1/2} \), where \( u = g(x) \)
- First, differentiate the outer function \( h(u) = u^{-1/2} \) with respect to \( u \), giving us \( h'(u) = -\frac{1}{2}u^{-3/2} \).
- Next, differentiate the inner function \( g(x) = \ln x \), which yields \( g'(x) = \frac{1}{x} \).
- Combine these derivatives, thanks to the chain rule: \(-\frac{1}{2}u^{-3/2} \cdot \frac{1}{x}\).
Differentiation Techniques
Differentiation is the process of finding the derivative of a function. It is the core method used in calculus to determine how a function changes at any given point. Differentiation allows us to solve many problems involving rates of change and slopes of curves.
In the problem \( f(x) = \frac{1}{\sqrt{\ln x}} \), various differentiation techniques come into play:
In the problem \( f(x) = \frac{1}{\sqrt{\ln x}} \), various differentiation techniques come into play:
- Power Rule: This rule is handy when you deal with polynomial expressions. Here, we rewrote \( \sqrt{\ln x} \) as \((\ln x)^{-1/2}\) to apply power differentiation rules.
- Logarithmic Differentiation: Given that \( \ln x \) is a key part of the function, its derivative \( \frac{1}{x} \) is crucial for the solution.
- Chain Rule: As previously mentioned, this is used when dealing with compositions of functions.
