Chain rule with the power rule

Khan Academy
22 Jul 201605:52
EducationalLearning
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TLDRThe video script presents a detailed explanation on how to find the derivative of a complex function using the chain rule. The function f(x) is broken down into two simpler functions, u(x) and v(x), where u(x) represents the initial operations and v(x) represents the final operation (raising to the eighth power). By applying the chain rule, the derivative of f(x) is calculated as the product of the derivative of v with respect to u (which is u(x) raised to the seventh power multiplied by 8) and the derivative of u with respect to x (which is 6x^2 + 10x). The video emphasizes the utility of the chain rule in simplifying the process of differentiating composite functions.

Takeaways
  • πŸ“š The function f(x) = 2x^3 + 5x^2 - 7 can be differentiated using the chain rule.
  • πŸ” The function is viewed as a composition of two functions, u(x) and v(x), where u(x) = 2x^3 + 5x^2 - 7 and v(u) = u^8.
  • 🌟 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.
  • πŸ“ˆ To find u'(x), apply the power rule to each term of u(x), resulting in u'(x) = 6x^2 + 10x.
  • πŸ“ˆ To find v'(u), apply the power rule to u^8, which gives v'(u) = 8u^7.
  • πŸ“ˆ The derivative of a constant is zero, so it does not affect the derivative of the function.
  • πŸ”’ The final expression for f'(x) is 8(2x^3 + 5x^2 - 7)^7 * (6x^2 + 10x).
  • πŸ’‘ With practice, the process of using the chain rule becomes more intuitive and can be done mentally without writing out all the steps.
  • πŸ“Š When differentiating composite functions, focus on the derivative of the outer function with respect to the inner function.
  • πŸ“Š The chain rule is a powerful tool for differentiating complex functions and is widely used in calculus.
Q & A

  • What is the given function f(x) in the script?

    -The given function f(x) is 2x^3 + 5x^2 - 7.

  • How is the function f(x) described as a composition of two functions?

    -The function f(x) is described as a composition of two functions, u(x) and v(x), where u(x) = 2x^3 + 5x^2 - 7 and v(x) is the eighth power of its input.

  • What is the role of the chain rule in finding the derivative of f(x)?

    -The chain rule is used to find the derivative of the composite function f(x) by multiplying the derivative of the outer function (v) with respect to the inner function (u) by the derivative of the inner function (u) with respect to x.

  • What is the derivative of u(x) with respect to x?

    -The derivative of u(x) with respect to x, denoted as u'(x), is 6x^2 + 10x.

  • What is the derivative of v(x) with respect to x?

    -The derivative of v(x) with respect to x, denoted as v'(x), is 8x^7.

  • How is v'(x) expressed in terms of u(x)?

    -v'(x) is expressed in terms of u(x) as 8 * (2x^3 + 5x^2 - 7)^7.

  • What is the final expression for the derivative f'(x)?

    -The final expression for the derivative f'(x) is 8 * (2x^3 + 5x^2 - 7)^7 * (6x^2 + 10x).

  • What is the significance of breaking down f(x) into u(x) and v(x)?

    -Breaking down f(x) into u(x) and v(x) allows us to apply the chain rule more effectively, simplifying the process of finding the derivative of the composite function.

  • How does the power rule assist in finding the derivatives of u(x) and v(x)?

    -The power rule is used to find the derivatives of u(x) and v(x) by applying the exponent decrease for derivatives, which results in u'(x) = 6x^2 + 10x and v'(x) = 8x^7.

  • What is the purpose of the chain rule in calculus?

    -The chain rule is used to find the derivative of a composite function by breaking it down into simpler functions and evaluating the derivative step by step, which simplifies the process of differentiation.

  • Why is it important to practice the chain rule?

    -Practicing the chain rule is important because it helps in understanding and efficiently calculating the derivatives of complex composite functions, which is crucial in various applications of calculus.

Outlines
00:00
πŸ“š Understanding the Chain Rule for Derivatives

The paragraph introduces the concept of finding the derivative of a complex function by breaking it down into simpler sub-functions. The function f(x) is given as 2x^3 + 5x^2 - 7, all raised to the eighth power. The voiceover explains that this can be viewed as a composition of two functions, u(x) and v(x), where u(x) represents the initial transformation of x, and v(x) represents the final transformation of u(x). The process of finding the derivative, denoted as f'(x), involves applying the chain rule, which 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. The paragraph provides a step-by-step explanation of how to calculate u(x), u'(x), v(x), and v'(x), and then how to combine these to find the derivative of the original function f(x). The final expression for f'(x) is given as 8 times the entire u(x) expression raised to the seventh power, multiplied by u'(x), which is 6x^2 + 10x.

05:01
🧠 Applying the Chain Rule for Practical Problem-Solving

This paragraph continues the discussion on the chain rule, emphasizing its practical application in solving derivative problems. It explains that with practice, one can quickly identify the outer and inner functions and apply the chain rule without having to write down every step. The paragraph clarifies that the derivative of the outer function with respect to the inner function is found by replacing the x in the outer function with the inner function, and then applying the power rule. In this case, the derivative of v(x) with respect to u(x) is 8 times u(x) to the seventh power. This is then multiplied by the derivative of the inner function, u'(x), which is 6x^2 + 10x, to obtain the final derivative of the composite function f(x).

Mindmap
Keywords
πŸ’‘Function
In mathematics, a function is a relation that assigns a single output value to each input value. In the context of the video, f(x) represents a function where the input is x and the output is calculated based on the given mathematical expression. The main theme of the video revolves around finding the derivative of this function, which is a process to determine how the function changes as its input changes.
πŸ’‘Derivative
The derivative of a function is a measure of the rate at which the function changes with respect to its input variable. It represents the slope of the tangent line to the graph of the function at any given point and is a fundamental concept in calculus. The video's main objective is to find the derivative of the given function f(x) with respect to x.
πŸ’‘Composition of Functions
The composition of functions is a mathematical operation where one function is applied to the result of another function. It is denoted by writing one function after another, like f(g(x)). In the video, the function f(x) is viewed as a composition of two functions, u(x) and v(x), which simplifies the process of finding the derivative using the chain rule.
πŸ’‘Chain Rule
The chain rule is a fundamental rule in calculus that is used to find the derivative of a composite function. It states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function. The video uses the chain rule to find the derivative of the function f(x) by breaking it down into simpler functions u(x) and v(x).
πŸ’‘Power Rule
The power rule is a basic differentiation rule in calculus that states that the derivative of x^n, where n is any real number, is n*x^(n-1). This rule is used in the video to find the derivatives of the functions u(x) and v(x) when finding the derivative of the composite function f(x).
πŸ’‘u(x)
In the video, u(x) is an intermediate function defined as u(x) = 2x^3 + 5x^2 - 7. It represents the first part of the composite function f(x) before it is input into the second function v(x). The derivative of u(x) with respect to x is calculated using the power rule and is essential for applying the chain rule to find the derivative of f(x).
πŸ’‘v(x)
v(x) is the second intermediate function in the composition of f(x), which takes the output of u(x) as its input and raises it to the eighth power. The function v(x) is crucial for understanding the composition and applying the chain rule to find the derivative of f(x).
πŸ’‘u'(x)
u'(x) represents the derivative of the intermediate function u(x) with respect to x. It is calculated using the power rule and is a key component in applying the chain rule to find the derivative of the composite function f(x).
πŸ’‘v'(u(x))
v'(u(x)) is the derivative of the intermediate function v(x) evaluated at the output of u(x). It represents how v(x) changes with respect to changes in u(x) and is used in the chain rule to find the derivative of the composite function f(x).
πŸ’‘f'(x)
f'(x) is the derivative of the composite function f(x) with respect to x. It is the final result of applying the chain rule and finding how f(x) changes as x changes. The calculation of f'(x) is the main goal of the video and demonstrates the application of the chain rule and the power rule.
πŸ’‘Slope of Tangent Line
The slope of the tangent line to a function at a particular point is a measure of how steep the graph is at that point. The derivative of a function at a point gives the slope of the tangent line at that point. In the video, the derivative f'(x) is being sought to understand the rate of change and the slope of the tangent line to the graph of the function f(x) at any given x-value.
Highlights

The function f(x) is described as f(x) = (2x^3 + 5x^2 - 7)^8.

The problem involves finding the derivative of the function f with respect to x.

The function can be viewed as a composition of two functions, u(x) and v(x).

u(x) is defined as 2x^3 + 5x^2 - 7.

v(x) is defined as taking the eighth power of its input.

f(x) can be expressed as v(u(x)).

The chain rule is key to solving this problem.

f'(x) is calculated as the product of v'(u(x)) and u'(x).

u'(x) is calculated using basic derivative properties and the power rule.

u'(x) is equal to 6x^2 + 10x.

v'(x) is calculated using the power rule, resulting in 8x^7.

v'(u(x)) is found by substituting u(x) into v'(x), resulting in 8(2x^3 + 5x^2 - 7)^7.

f'(x) is the product of v'(u(x)) and u'(x), giving the final derivative expression.

The derivative f'(x) is 8(2x^3 + 5x^2 - 7)^7 multiplied by (6x^2 + 10x).

With practice, the chain rule can be applied more efficiently to solve such problems.

The process simplifies to taking the derivative of the outer function with respect to the inner function, and then multiplying by the derivative of the inner function.

This method can be applied to more complex functions and demonstrates the power of the chain rule in calculus.

Transcripts

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