Worked example: Derivative of cos_(x) using the chain rule | AP Calculus AB | Khan Academy
TLDRThe video script discusses the application of the chain rule in calculus to find the derivative of a composite function, specifically focusing on the function f(x) = cos(x)^3. The explanation is detailed, walking through the process of differentiating the outer function with respect to the inner function and then differentiating the inner function with respect to x. The final result is the derivative f'(x) = -3sin(x)cos^2(x), highlighting the chain rule's utility in understanding and solving complex calculus problems.
Takeaways
- π The function under consideration is f(x) = cos(x)^3, which is a composite function.
- π§ To find the derivative of f(x), the chain rule is essential and will be applied.
- π The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
- π The derivative of the outer function (v) with respect to the inner function (u) is v'(u) = 3u^2, since v(u) = u^3.
- π The derivative of the inner function (u) with respect to x is u'(x) = -sin(x), since u(x) = cos(x).
- π’ The derivative of f(x) is found by multiplying the derivative of v with respect to u by the derivative of u with respect to x.
- π This results in f'(x) = 3(cos(x)^2)(-sin(x)), which simplifies to -3cos^2(x)sin(x).
- π‘ Understanding the chain rule is crucial for differentiating composite functions and is a fundamental concept in calculus.
- π The process of differentiating composite functions involves recognizing the inner and outer functions and applying the chain rule accordingly.
- π The explanation provided in the script helps deepen the understanding of the chain rule and its application in various calculus problems.
Q & A
What is the function f(x) described in the transcript?
-The function f(x) described in the transcript is f(x) = cos(x)^3, which is the cosine of x raised to the third power.
What mathematical concept is being used to find f'(x)?
-The chain rule is being used to find the derivative f'(x) of the function f(x).
How is the composite function represented in the transcript?
-The composite function is represented as f(x) = v(u(x)), where u(x) = cos(x) and v(u) = u^3.
What is the derivative of v with respect to u in the given context?
-The derivative of v with respect to u is 3u^2, since v(u) = u^3.
What is the derivative of u with respect to x?
-The derivative of u with respect to x is the derivative of cos(x), which is -sin(x).
What is the final expression for f'(x) according to the transcript?
-The final expression for f'(x) is -3sin(x)cos^2(x).
How does the chain rule help in understanding the derivative of composite functions?
-The chain rule helps in understanding the derivative of composite functions by breaking down the composite function into simpler functions and applying the derivative to each part, multiplying them together to get the final derivative.
What is the significance of the chain rule in calculus?
-The chain rule is significant in calculus as it allows the calculation of derivatives for composite functions, which are common in various mathematical and real-world applications.
How does the transcript explain the concept of derivatives in the context of the chain rule?
-The transcript explains the concept of derivatives in the context of the chain rule by using a step-by-step approach, first identifying the inner and outer functions, calculating their derivatives separately, and then applying the chain rule to combine these results.
What is the general form of the chain rule as described in the transcript?
-The general form of the chain rule as described in the transcript is (d/dx)[f(g(x))] = f'(g(x)) * g'(x).
How does the transcript illustrate the process of applying the chain rule?
-The transcript illustrates the process of applying the chain rule by first identifying the composite function, defining the inner function u(x) and the outer function v(u), calculating their derivatives separately, and then multiplying these derivatives to find the derivative of the entire composite function with respect to x.
Outlines
π Introduction to the Chain Rule and Composite Functions
This paragraph introduces the concept of the chain rule in calculus, specifically in the context of finding the derivative of a composite function. The function f(x) = cos(x)^3 is used as an example to demonstrate how the chain rule can be applied. The explanation emphasizes the importance of understanding the relationship between composite functions and their derivatives, and it sets the stage for a deeper exploration of the chain rule and its applications in calculus.
π’ Derivative Calculation using the Chain Rule
In this paragraph, the video script delves into the actual calculation of the derivative of the given function f(x) = cos(x)^3 using the chain rule. It explains the process of taking the derivative of the outer function with respect to the inner function (v' cos(x) * u') and then multiplying it by the derivative of the inner function with respect to x (cos(x)). The result is a detailed step-by-step breakdown of the chain rule application, leading to the final derivative f'(x) = -3cos^2(x)sin(x). This comprehensive explanation helps viewers understand not only the 'how' but also the 'why' behind the chain rule.
Mindmap
Keywords
π‘Function
π‘Derivative
π‘Chain Rule
π‘Composite Function
π‘Trigonometric Function
π‘Power Function
π‘Sine Function
π‘Exponential Function
π‘Differential Notation
π‘Negative Sine
Highlights
The function f(x) is defined as the cosine of x to the third power.
The chain rule is essential for finding the derivative of the given function.
The function is a composite function, with an inner function (cosine of x) and an outer function (raising to the third power).
The chain rule connects the derivative of a composite function to the derivatives of its components.
The derivative of f(x) with respect to x can be found by applying the chain rule as v'(u(x)) * u'(x).
The derivative of the outer function v, which is raising to the third power, is simply 3 times the inner function squared.
The inner function u(x) is the cosine of x.
The derivative of the inner function u with respect to x is the negative sine of x.
The derivative of the outer function with respect to the inner function is three times the inner function squared.
The final derivative of f(x) with respect to x is negative three times the sine of x times the cosine of x squared.
The process of finding the derivative involves understanding the chain rule and applying it to composite functions.
The explanation emphasizes the importance of visualizing the composite function and its components.
The chain rule is a fundamental concept in calculus with wide applications in various mathematical problems.
The transcript provides a detailed step-by-step breakdown of the chain rule application.
The method can be generalized to find derivatives of composite functions involving different inner and outer functions.
The explanation aims to deepen the understanding of the chain rule and its application in calculus.
Transcripts
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