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By the end of this lesson, students will be able to solve related rates problems by applying the chain rule and implicit differentiation to real-world scenarios.

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Students will:
- Set up equations relating multiple variables
- Apply implicit differentiation with respect to time
- Interpret rates of change in context
- Solve practical problems involving related rates

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How can we use calculus to model and solve real-world problems where multiple quantities change simultaneously?

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Real-world example: A conical water tank is being filled. How does the height of water relate to its volume over time?

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Teacher demonstrates the steps for solving related rates:
1. Draw a diagram
2. List known variables and rates
3. Write an equation relating variables
4. Differentiate with respect to time
5. Substitute known values and solve

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Rates of change between radius and area of circle

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Students collaborate on Padlet to solve: A ladder sliding down a wall. Share diagrams and solution steps with class discussion.

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Work through these GeoGebra simulations to visualize related rates problems

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Contains Dynamic Illustrations depicting related-rates problems often seen in a Caluclus-1 course.

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Advanced students: Investigate related rates in optimization problems involving maximum/minimum scenarios.

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Review of chain rule and implicit differentiation with additional practice problems

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Section 3.10 : Implicit Differentiation

For problems 1 - 3 do each of the following.

1. Find \(y'\) by solving the equation for y and differentiating directly.

2. Find \(y'\) by implicit differentiation.

3. Check that the derivatives in (a) and (b) are the same.

For problems 4 - 9 find \(y'\) by implicit differentiation.

1. \(2{y^3} + 4{x^2} - y = {x^6}\) Solution

2. \(7{y^2} + \sin \left( {3x} \right) = 12 - {y^4}\) Solution

3. \({{\bf{e}}^x} - \sin \left( y \right) = x\) Solution

4. \(4{x^2}{y^7} - 2x = {x^5} + 4{y^3}\) Solution

5. \(\cos \left( {{x^2} + 2y} \right) + x\,{{\bf{e}}^{{y^{\,2}}}} = 1\) Solution

6. \(\tan \left( {{x^2}{y^4}} \right) = 3x + {y^2}\) Solution

For problems 10 & 11 find the equation of the tangent line at the given point.

1. \({x^4} + {y^2} = 3\) at \(\left( {1,\, - \sqrt 2 } \right)\). Solution

2. \({y^2}{{\bf{e}}^{2x}} = 3y + {x^2}\) at \(\left( {0,3} \right)\). Solution

For problems 12 & 13 assume that \(x = x\left( t \right)\), \(y = y\left( t \right)\) and \(z = z\left( t \right)\) then differentiate the given equation with respect to t.

1. \({x^2} - {y^3} + {z^4} = 1\) Solution

2. \({x^2}\cos \left( y \right) = \sin \left( {{y^3} + 4z} \right)\) Solution

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Calculus I - Implicit Differentiation (Practice Problems)

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Here is a set of practice problems to accompany the Implicit Differentiation section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University.

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Students share their work on a collaborative Padlet board, explaining their problem-solving process for peer review.

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Three related rates problems of increasing difficulty to assess understanding.

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Calculus I - Related Rates (Practice Problems)

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Here is a set of practice problems to accompany the Related Rates section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University.

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In pairs, students create and solve their own related rates problem, presenting their work to the class.

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Class discussion: What are the key steps in solving any related rates problem? What common mistakes should we avoid?

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Individual reflection:
1. What was the most challenging part of today's lesson?
2. How does related rates connect to real-world applications?
3. What questions do you still have?

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Complete 5 related rates problems from the provided worksheet, showing all steps clearly.

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Find and analyze a real-world scenario that involves related rates. Create a brief video explanation to share in next class.

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Complete the interactive exercises on related rates