Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
14. y = ln(2θ+2)
11
views
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
1:38 minutesProblem 7.2.25
Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
25. y = ln(ln(x))
Verified step by step guidance1
Identify the function given: \(y = \ln(\ln(x))\). This is a composition of two functions: the natural logarithm of another natural logarithm.
Apply the chain rule for derivatives. The chain rule states that if \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\).
Set the outer function as \(f(u) = \ln(u)\) and the inner function as \(u = \ln(x)\). Then, find the derivative of the outer function with respect to \(u\): \(f'(u) = \frac{1}{u}\).
Find the derivative of the inner function \(u = \ln(x)\) with respect to \(x\): \(u' = \frac{1}{x}\).
Combine these results using the chain rule: \(\frac{dy}{dx} = \frac{1}{\ln(x)} \cdot \frac{1}{x}\). This expresses the derivative of \(y\) with respect to \(x\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chain Rule
The chain rule is a fundamental technique for finding the derivative of composite functions. It states that the derivative of a function composed of two functions is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
Recommended video:
05:02
Intro to the Chain Rule
Derivative of the Natural Logarithm Function
The derivative of the natural logarithm function ln(x) with respect to x is 1/x, provided x > 0. This rule is essential when differentiating expressions involving logarithms, especially nested logarithms like ln(ln(x)).
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Domain Considerations for Logarithmic Functions
When differentiating logarithmic functions, it's important to consider the domain where the function is defined. For y = ln(ln(x)), both ln(x) and x must be positive, so x > 1. This ensures the function and its derivative are valid and real-valued.
Recommended video:
5:26Graphs of Logarithmic Functions
Related Videos
Related Practice