Derivative of sin(ln(x_)) | Advanced derivatives | AP Calculus AB | Khan Academy
TLDRThe video script presents a step-by-step guide on how to derive the compound function of the sine of the natural log of x squared. It simplifies the process by breaking down the compound function into its constituent functions (f(x)=sine(x), g(x)=ln(x), and h(x)=x^2) and applying the chain rule iteratively. The explanation walks through the derivatives of the outer function with respect to the inner functions, leading to the final simplified result of 2/x times the cosine of the natural log of x squared. The analogy of peeling an onion is used to illustrate the layer-by-layer approach to solving the derivative, emphasizing the method's clarity and manageability.
Takeaways
- 📚 The problem involves taking the derivative of a composite function, specifically the sine of the natural log of x squared.
- 🔄 The approach is to break down the composite function into simpler parts: f(x) = sine of x, g(x) = natural log of x, and h(x) = x squared.
- 🌀 The process starts by identifying the outer function (f) and taking its derivative with respect to the next level of composition (g of h of x).
- 📈 The derivative of sine x is cosine x, which will be applied to the inner function (natural log of x squared).
- 🏄♂️ The next step is to take the derivative of the inner function (g) with respect to its input (x squared), which involves the natural log function.
- 🔢 For the natural log function, the derivative is 1/x, but since we have x squared, it becomes 1/(x squared).
- 🔧 The final step is to find the derivative of the innermost function (h), which is x squared, resulting in 2x.
- 📊 By applying the chain rule repeatedly, we multiply the derivatives of the individual functions together, accounting for the composition.
- 🧩 The process is likened to peeling an onion, where each layer (each function in the composition) is addressed in turn.
- 🎯 The final result simplifies to 2/x times the cosine of the natural log of x squared, showcasing how complex derivatives can be broken down and solved step by step.
- 💡 This example serves as a clear demonstration of the chain rule in action and the importance of understanding the individual components of a composite function when taking derivatives.
Q & A
What is the main topic of the video script?
-The main topic of the video script is the process of finding the derivative of the composite function, specifically the sine of the natural log of x squared.
How does the script introduce the composite function?
-The script introduces the composite function by breaking it down into three components: f(x) as the sine of x, g(x) as the natural log of x, and h(x) as x squared.
What is the first step in differentiating the composite function?
-The first step in differentiating the composite function is to find the derivative of the outermost function, f(x), with respect to the next level of composition, which is g(h(x)).
What is the derivative of sine of x?
-The derivative of sine of x is cosine of x.
How does the script handle the inner composition of natural log and x squared?
-The script handles the inner composition by treating it as g(x) and applying the chain rule. The derivative of natural log of x (ln x) is 1/x, and since we have x squared, it becomes 1/(x squared).
What is the final derivative of the composite function?
-The final derivative of the composite function is 2 over x times the cosine of the natural log of x squared, after simplifying the expression by canceling out x squared.
What is the method used in the script to simplify the expression?
-The method used to simplify the expression is by changing the order of multiplication and canceling out terms, such as x squared in the numerator and denominator.
How does the script describe the process of differentiating composite functions?
-The script describes the process as similar to peeling an onion, where you start with the outermost function and work your way inwards, applying the chain rule at each level of composition.
What is the derivative of x squared with respect to x?
-The derivative of x squared with respect to x is 2x.
How does the script emphasize the importance of understanding the chain rule in this context?
-The script emphasizes the importance of understanding the chain rule by showing that it is necessary to apply it multiple times when dealing with composite functions, and it helps in breaking down complex derivatives into simpler, manageable parts.
What is the significance of the color-coding used in the script?
-The color-coding used in the script is meant to visually differentiate the various components and intermediate steps of the derivative process, making it easier for the viewer to follow along and understand the progression.
Outlines
📚 Derivative of Composite Functions
This paragraph discusses the process of finding the derivative of a composite function, specifically the sine of the natural log of x squared. The explanation begins by breaking down the composite function into its constituent functions: f(x) as sine of x, g(x) as the natural log of x, and h(x) as x squared. The paragraph then illustrates the method of taking the derivative of the outer function with respect to the immediate inner function, leading to the application of the chain rule. The explanation continues with the differentiation of the inner functions, emphasizing the importance of understanding the derivative of sine, the natural log function, and the innermost function (x squared). The paragraph concludes by simplifying the expression to 2 over x times the cosine of the natural log of x squared, highlighting the step-by-step approach to tackling complex derivatives.
Mindmap
Keywords
💡derivative
💡composite function
💡chain rule
💡sine
💡natural log
💡x squared
💡cosine
💡1/x
💡2x
💡cancellation
💡onion analogy
Highlights
Attempting to take the derivative of the sine of the natural log of x squared.
Composite function of multiple layers: f(x) = sine of x, g(x) = natural log of x, and h(x) = x squared.
Derivative of the outer function f with respect to the next level of composition results in cosine of the inner function.
Derivative of sine of x is cosine of x, but applied to the inner function, which is the natural log of x squared.
The derivative process involves applying the chain rule multiple times due to the composite nature of the function.
Derivative of natural log (ln) of x is 1/x, but applied to x squared, it becomes 1/(x squared).
Innermost function derivative: the derivative of x squared with respect to x is 2x.
The process of finding the derivative is likened to peeling an onion, layer by layer.
Simplification of the expression by changing the order of multiplication and canceling out terms.
The final simplified derivative is 2 over x times the cosine of the natural log of x squared.
The approach to solving the derivative is to first address the outer function and then proceed inwards.
The problem, though seemingly complex, is broken down into manageable steps using the chain rule.
Each layer of the composite function is addressed individually, highlighting the importance of understanding the structure of composite functions.
The explanation emphasizes the method of thinking through the derivative process without necessarily writing out all the mathematical notation.
The final result is a clear and concise expression that demonstrates the utility of the chain rule in derivative calculations.
Transcripts
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