Implicit Differentiation | MIT 18.01SC Single Variable Calculus, Fall 2010

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7 Jan 201108:16
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TLDRIn this recitation, the professor introduces the concept of implicit differentiation with an example involving the curve defined by y^3 + x^3 = 3xy. The challenge is to find the slope of the tangent line at the point (4/3, 2/3). The professor emphasizes the difficulty of solving for y explicitly and instead demonstrates the use of implicit differentiation. After differentiating both sides of the equation with respect to x, the professor simplifies to isolate dy/dx, resulting in a formula for the slope of the tangent line at any point on the curve. The final step is to substitute the given point into the formula to find the slope, which is calculated to be 5/4, slightly greater than 1.

Takeaways
  • 📚 The lecture focuses on implicit differentiation, specifically to compute derivatives of functions defined implicitly.
  • 🔍 The example curve is defined by the equation y^3 + x^3 = 3xy.
  • 📝 The goal is to find the slope of the tangent line to the curve at the point (4/3, 2/3).
  • ✅ The point (4/3, 2/3) is confirmed to be on the curve by plugging in the values and verifying the equation.
  • 🔑 Implicit differentiation is preferred over solving explicitly for y, as the latter is complex and beyond the course scope.
  • 📏 To find the derivative implicitly, apply the chain rule to y^3 and the product rule to 3xy.
  • ➕ The derivative of y^3 is 3y^2(dy/dx) and the derivative of x^3 is 3x^2.
  • 🧮 For the right side of the equation, the product rule gives 3y + 3x(dy/dx).
  • ⚖️ Combine like terms to isolate dy/dx, resulting in dy/dx = (y - x^2) / (y^2 - x).
  • 🔢 Plug in the specific point (4/3, 2/3) to find the slope at that point, which simplifies to 5/4.
  • 🎯 The computed slope of 5/4 is reasonable and aligns with the graphical representation of the tangent line.
Q & A

  • What is the main mathematical concept discussed in the transcript?

    -The main concept discussed is implicit differentiation, which is used to find the derivative of functions that are defined implicitly rather than explicitly.

  • What is the implicit equation given in the example?

    -The implicit equation given is y^3 + x^3 = 3xy.

  • Why is it difficult to solve the equation explicitly for y?

    -It is difficult to solve the equation explicitly for y because it involves both x and y in a non-linear way, making it challenging to isolate y on one side of the equation.

  • What is the purpose of finding the derivative in this problem?

    -The purpose of finding the derivative is to determine the slope of the tangent line to the curve at a specific point, (4/3, 2/3).

  • Which differentiation rules are mentioned as being used in this problem?

    -The chain rule and the product rule are mentioned as being used to differentiate the given implicit equation.

  • How is the derivative of y^3 with respect to x calculated?

    -The derivative of y^3 with respect to x is calculated using the chain rule, resulting in 3y^2 * (dy/dx).

  • What is the resulting expression for dy/dx after applying implicit differentiation?

    -The resulting expression for dy/dx is (y - x^2) / (y^2 - x).

  • How is the slope of the tangent line at the point (4/3, 2/3) determined?

    -The slope is determined by plugging the values x = 4/3 and y = 2/3 into the expression for dy/dx, yielding a slope of 5/4.

  • Why does the professor suggest using implicit differentiation instead of solving for y explicitly?

    -The professor suggests using implicit differentiation because solving for y explicitly is complex and beyond the scope of the course, whereas implicit differentiation is more straightforward in this context.

  • What conclusion is drawn about the slope of the tangent line at the end of the problem?

    -The conclusion is that the slope of the tangent line at the point (4/3, 2/3) is 5/4, and this value is consistent with the professor's rough sketch of the curve.

Outlines
00:00
📘 Introduction to Implicit Differentiation

The professor welcomes students back and recaps the previous lecture on implicit differentiation. He introduces the problem for the day, which involves finding the slope of the tangent line to the curve defined by \( y^3 + x^3 = 3xy \) at the point \((4/3, 2/3)\). He explains that verifying the point lies on the curve is possible through substitution and emphasizes the complexity of solving for \( y \) explicitly. He suggests that implicit differentiation is a better approach for this problem and encourages students to attempt solving it themselves before proceeding.

05:01
📝 Starting Implicit Differentiation

The professor begins solving the problem by using implicit differentiation. He explains that the chain rule is essential because \( y \) is implicitly a function of \( x \). He derives the expression for the derivative of \( y^3 \) with respect to \( x \), using the chain rule to get \( 3y^2 \cdot \frac{dy}{dx} \). He then differentiates \( x^3 \) straightforwardly to get \( 3x^2 \). On the right side of the equation, he identifies a product rule situation for \( 3xy \) and differentiates accordingly, resulting in \( 3y + 3x \cdot \frac{dy}{dx} \). This step-by-step differentiation process demonstrates the application of the chain and product rules.

Mindmap
Keywords
💡Implicit Differentiation
Implicit differentiation is a method used to find the derivative of a function that is not explicitly expressed in terms of a single variable. In the video, it is the central technique for finding the slope of the tangent line to a curve defined by an equation that does not isolate y in terms of x. The professor discusses using implicit differentiation to compute derivatives when explicit differentiation is too complex or infeasible.
💡Derivative
A derivative in calculus represents the rate at which a function changes with respect to its variable. In the context of the video, the derivative is essential for determining the slope of the tangent line to the curve at a specific point, which is a fundamental concept in understanding how the curve behaves locally.
💡Tangent Line
A tangent line is a straight line that touches a curve at a single point without crossing it. The slope of this line at any given point on the curve is the derivative of the curve at that point. In the video, the professor is interested in finding the slope of the tangent line to a specific curve at the point (4/3, 2/3).
💡Chain Rule
The chain rule is a fundamental principle in calculus for differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In the script, the chain rule is used to differentiate y^3 with respect to x, which is a key step in implicit differentiation.
💡Product Rule
The product rule is a formula used to find the derivative of a product of two functions. It states that the derivative of two functions multiplied together is the derivative of the first times the second plus the first times the derivative of the second. In the video, the product rule is applied to differentiate the term 3xy within the given equation.
💡Slope
Slope is a measure of the steepness of a line, and in the context of calculus, it is the value of the derivative of a curve at a particular point. The professor aims to find the slope of the tangent line to the curve at the point (4/3, 2/3), which is represented by the value of y' at that point.
💡Cubed Function
A cubed function is a mathematical function where the variable is raised to the power of three. In the video, both y^3 and x^3 are cubed functions, and the process of differentiating these terms with respect to x is part of the implicit differentiation process.
💡Rational Number Arithmetic
Rational number arithmetic refers to the operations performed on rational numbers, which are numbers that can be expressed as the quotient of two integers. In the video, the professor performs rational number arithmetic when simplifying the expression for the slope of the tangent line at the given point.
💡Differentiation Rules
Differentiation rules are the mathematical principles used to find the derivative of a function. These include the power rule, product rule, quotient rule, and chain rule. The video script refers to these rules when discussing the process of differentiating the given equation to find the slope of the tangent line.
💡Composite Functions
Composite functions are functions that are formed by applying one function to the result of another. In the script, when differentiating y^3 with respect to x, the function y is considered as a composite function of x, and the chain rule is applied to find its derivative.
💡Solving for y
Solving for y typically means expressing y in terms of x, which allows for explicit differentiation. The professor mentions that solving for y in the given equation is very difficult and not within the scope of the course, hence the preference for implicit differentiation.
Highlights

Introduction to implicit differentiation and its application in computing derivatives of implicitly defined functions.

Problem statement: Find the slope of the tangent line to the curve defined by y^3 + x^3 = 3xy at the point (4/3, 2/3).

Verification of the point (4/3, 2/3) being on the curve by plugging in the values.

Observation that solving for y explicitly is difficult and beyond the scope of the course.

Advantage of using implicit differentiation over explicit differentiation for this problem.

Starting the process of implicit differentiation by taking the derivative of both sides of the equation with respect to x.

Application of the chain rule to differentiate y^3 with respect to x.

Differentiation of x^3 as a straightforward process yielding 3x^2.

Use of the product rule for differentiating the term 3xy.

Isolating dy/dx by combining terms and simplifying the equation.

General expression for the slope of the tangent line at any point (x, y) on the curve.

Substitution of the specific point (4/3, 2/3) into the derived expression to find dy/dx.

Performing rational number arithmetic to simplify the expression for the slope at the given point.

Final calculation of the slope as 5/4, indicating a slope slightly greater than 1.

Visual confirmation of the calculated slope with the tangent line on the curve graph.

Transcripts

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