📐 Concept diagram

04-06 - Implicit Differentiation

Phase: 4 | Subject: 04-06 Prerequisites: 04-05-derivatives-of-elementary-functions.md Next subject: 04-07-applications-of-derivatives.md

Learning Objectives

By the end of this subject, you will be able to:

1. Differentiate implicitly defined functions
2. Use implicit differentiation to find dy/dx
3. Find equations of tangent lines to implicit curves
4. Apply logarithmic differentiation
5. Solve related rates problems

Core Content

Explicit vs Implicit

Explicit: y = f(x) — y is isolated Implicit: F(x, y) = 0 — x and y mixed together

Example explicit: y = x² + 3 Example implicit: x² + y² = 25 (circle)

For implicit functions, we differentiate BOTH SIDES with respect to x, treating y as a function of x.

Implicit Differentiation Steps

1. Differentiate both sides with respect to x
2. For terms with y, use chain rule: d/dx[yⁿ] = n·y^(n-1)·dy/dx
3. Collect all dy/dx terms on one side
4. Factor out dy/dx
5. Solve for dy/dx

Example: x² + y² = 25

1. d/dx[x²] + d/dx[y²] = d/dx[25]
2. 2x + 2y·dy/dx = 0
3. 2y·dy/dx = -2x
4. dy/dx = -x/y

Interpretation: At point (3, 4): dy/dx = -3/4. The tangent slope is -3/4.

Logarithmic Differentiation

Useful for products, quotients, or powers with variables in both base and exponent.

Steps: 1. Take natural log of both sides 2. Use log laws to simplify 3. Differentiate implicitly 4. Solve for dy/dx

Example: y = xˣ

1. ln(y) = ln(xˣ) = x·ln(x)
2. d/dx[ln(y)] = d/dx[x·ln(x)]
3. (1/y)·dy/dx = 1·ln(x) + x·(1/x) = ln(x) + 1
4. dy/dx = y·(ln(x) + 1) = xˣ(ln(x) + 1)

Related Rates

⚠️ THIS IS CRITICAL — related rates problems appear throughout physics, engineering, and economics. The key insight is that when quantities are related by an equation, their rates of change are related by differentiation.

General approach: 1. Identify what is given (known rates) and what is asked (unknown rate) 2. Write an equation relating the quantities (not the rates — those come from differentiation) 3. Differentiate both sides with respect to TIME (or the relevant variable) 4. Substitute known values and solve for the unknown rate

Example: A spherical balloon is being filled with air at a rate of 100 cm³/s. How fast is the radius increasing when the radius is 5 cm?

1. Known: dV/dt = 100 cm³/s. Unknown: dr/dt when r = 5.
2. Volume of sphere: V = (4/3)πr³
3. Differentiate with respect to t: dV/dt = (4/3)π · 3r² · dr/dt = 4πr² · dr/dt
4. Substitute: 100 = 4π(5)² · dr/dt = 100π · dr/dt
5. dr/dt = 100/(100π) = 1/π ≈ 0.318 cm/s

Example: A 5-metre ladder slides down a wall. The bottom moves away from the wall at 2 m/s. How fast is the top sliding down when the bottom is 3 m from the wall?

1. Let x = distance from wall to bottom, y = height of top.
2. Relation: x² + y² = 25 (Pythagoras on 5-m ladder)
3. Differentiate: 2x·dx/dt + 2y·dy/dt = 0
4. Known: dx/dt = 2, x = 3. Find y: 3² + y² = 25 → y = 4.
5. 2(3)(2) + 2(4)·dy/dt = 0 → 12 + 8·dy/dt = 0 → dy/dt = -12/8 = -3/2 = -1.5 m/s
6. The top slides DOWN at 1.5 m/s (negative sign means decreasing height).

Key Terms

• 04 06 Implicit Differentiation
• Correct: A)
• Correct: B)
• Example 1: Circle
• Example 2: Product
• Example 3: Logarithmic differentiation
• Explicit vs Implicit
• Implicit Differentiation Steps
• Logarithmic Differentiation
• Related Rates

Worked Examples

Example 1: Circle

x² + y² = 16. Find dy/dx and tangent at (2, 2√3).

1. 2x + 2y·dy/dx = 0
2. dy/dx = -x/y
3. At (2, 2√3): dy/dx = -2/(2√3) = -1/√3 = -√3/3

Example 2: Product

x·y + x - y = 5. Find dy/dx.

1. d/dx[xy] + d/dx[x] - d/dx[y] = 0
2. y + x·dy/dx + 1 - dy/dx = 0 (product rule on xy)
3. x·dy/dx - dy/dx = -y - 1
4. dy/dx(x - 1) = -(y + 1)
5. dy/dx = -(y + 1)/(x - 1)

Example 3: Logarithmic differentiation

y = (x² + 1)³/√(x - 1). Find dy/dx.

1. Take ln of both sides: ln(y) = 3·ln(x² + 1) - (1/2)·ln(x - 1)
2. Differentiate: (1/y)·dy/dx = 3·(2x)/(x² + 1) - (1/2)·(1/(x - 1))
3. dy/dx = y · [6x/(x² + 1) - 1/(2(x - 1))]
4. Substitute y: dy/dx = (x² + 1)³/√(x - 1) · [6x/(x² + 1) - 1/(2(x - 1))]

Quiz

Q1: What does the concept of Explicit vs Implicit primarily refer to in this subject?

A) A computational error related to Explicit vs Implicit B) A historical anecdote about Explicit vs Implicit C) A visual representation of Explicit vs Implicit D) The definition and application of Explicit vs Implicit

Correct: D)

• If you chose A: This is incorrect. Explicit vs Implicit is defined as: the definition and application of explicit vs implicit. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Explicit vs Implicit is defined as: the definition and application of explicit vs implicit. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Explicit vs Implicit is defined as: the definition and application of explicit vs implicit. The other options describe different aspects that are not the primary focus.
• If you chose D: Explicit vs Implicit is defined as: the definition and application of explicit vs implicit. The other options describe different aspects that are not the primary focus. Correct!

Q2: What is the primary purpose of Implicit Differentiation Steps?

A) It is used only in advanced research contexts B) It replaces all other methods in this domain C) It is used to implicit differentiation steps in mathematical analysis D) It is primarily a historical notation system

Correct: C)

• If you chose A: This is incorrect. Implicit Differentiation Steps serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Implicit Differentiation Steps serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: Implicit Differentiation Steps serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose D: This is incorrect. Implicit Differentiation Steps serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Logarithmic Differentiation is TRUE?

A) Logarithmic Differentiation is an advanced topic beyond this subject's scope B) Logarithmic Differentiation is not related to this subject C) Logarithmic Differentiation is mentioned only as a historical footnote D) Logarithmic Differentiation is a fundamental concept covered in this subject

Correct: D)

• If you chose A: This is incorrect. Logarithmic Differentiation is a fundamental concept covered in this subject. This subject covers Logarithmic Differentiation as part of its core content.
• If you chose B: This is incorrect. Logarithmic Differentiation is a fundamental concept covered in this subject. This subject covers Logarithmic Differentiation as part of its core content.
• If you chose C: This is incorrect. Logarithmic Differentiation is a fundamental concept covered in this subject. This subject covers Logarithmic Differentiation as part of its core content.
• If you chose D: Logarithmic Differentiation is a fundamental concept covered in this subject. This subject covers Logarithmic Differentiation as part of its core content. Correct!

Q4: Based on the worked examples in this subject, what is the correct result?

A) The inverse of the correct answer B) A different result from a common mistake C) -x/y D) An unrelated numerical value

Correct: C)

• If you chose A: This is incorrect. The worked examples show that the result is -x/y. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is -x/y. The other options represent common errors.
• If you chose C: The worked examples show that the result is -x/y. The other options represent common errors. Correct!
• If you chose D: This is incorrect. The worked examples show that the result is -x/y. The other options represent common errors.

Q5: How are Logarithmic Differentiation and Related Rates related?

A) Logarithmic Differentiation and Related Rates are completely unrelated topics B) Logarithmic Differentiation is the inverse of Related Rates C) Logarithmic Differentiation and Related Rates are closely related concepts D) Logarithmic Differentiation is a special case of Related Rates

Correct: C)

• If you chose A: This is incorrect. Both Logarithmic Differentiation and Related Rates are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Logarithmic Differentiation and Related Rates are covered in this subject as interconnected topics.
• If you chose C: Both Logarithmic Differentiation and Related Rates are covered in this subject as interconnected topics. Correct!
• If you chose D: This is incorrect. Both Logarithmic Differentiation and Related Rates are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Example 1: Circle?

A) The main error with Example 1: Circle is using it when it is not needed B) Example 1: Circle has no common misconceptions C) A common mistake is confusing Example 1: Circle with a similar concept D) Example 1: Circle is always computed the same way in all contexts

Correct: C)

• If you chose A: This is incorrect. Students often confuse Example 1: Circle with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse Example 1: Circle with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: Students often confuse Example 1: Circle with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose D: This is incorrect. Students often confuse Example 1: Circle with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Example 2: Product?

A) Use Example 2: Product only in pure mathematics contexts B) Example 2: Product is not practically useful C) Avoid Example 2: Product unless explicitly instructed D) Apply Example 2: Product to solve problems in this subject's domain

Correct: D)

• If you chose A: This is incorrect. Example 2: Product is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Example 2: Product is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Example 2: Product is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: Example 2: Product is a practical tool used throughout this subject to solve relevant problems. Correct!

Practice Problems

1. Find dy/dx for x² + y² = 9 Answer: 2x + 2y·dy/dx = 0, so dy/dx = -x/y

2. Find dy/dx for y³ + xy = 1 Answer: 3y²·dy/dx + y + x·dy/dx = 0, dy/dx(3y² + x) = -y, dy/dx = -y/(3y² + x)

3. Tangent slope to x² + y² = 25 at (3, 4) Answer: dy/dx = -x/y = -3/4

4. Find dy/dx for y = xˣ using logarithmic differentiation Answer: ln(y) = x·ln(x). (1/y)y' = ln(x) + 1. y' = y(ln(x) + 1) = xˣ(ln(x) + 1)

5. Find dy/dx for (x + y)² = x - y Answer: 2(x + y)(1 + y') = 1 - y'. 2(x + y) + 2(x + y)y' = 1 - y'. y'(2x + 2y + 1) = 1 - 2x - 2y. y' = (1 - 2x - 2y)/(2x + 2y + 1).

6. Water pours into a conical tank (height 10 m, radius 4 m at top) at 2 m³/min. How fast is the water level rising when the water is 5 m deep? (Cone volume: V = (1/3)πr²h. By similar triangles: r/h = 4/10 = 2/5.) Answer: V = (1/3)π(2h/5)²h = (4π/75)h³. dV/dt = (4π/25)h²·dh/dt. 2 = (4π/25)(25)·dh/dt = 4π·dh/dt. dh/dt = 1/(2π) ≈ 0.159 m/min.

Summary

Key takeaways:

• Implicit differentiation: differentiate both sides, y' appears via chain rule
• d/dx[yⁿ] = n·y^(n-1)·dy/dx
• Collect y' terms, factor, solve for y'
• Logarithmic differentiation: ln both sides first, then differentiate
• Useful for y = xˣ, products/quotients of many functions

Pitfalls

• Forgetting the chain rule on y-terms. When differentiating y², you must write 2y·dy/dx, not just 2y. Every y-term produces a dy/dx factor because y is a function of x.
• Trying to solve for y first. A common reflex is to isolate y before differentiating (e.g., y = ±√(25-x²) for a circle). This is unnecessary and often harder — implicit differentiation handles the equation as-is.
• Missing product rule on mixed terms. For xy, d/dx[xy] = y + x·dy/dx (product rule). It's not just x·dy/dx or just y — both parts are needed.
• Neglecting to collect dy/dx terms. After differentiating, make sure all terms with dy/dx land on one side before factoring. Scattered dy/dx terms make solving messy.
• Forgetting the y'/y step in logarithmic differentiation. d/dx[ln(y)] = y'/y, not 1/y. The chain rule applies: derivative of outer ln gives 1/y, times inner derivative y'.

Next Steps

Next up: 04-07-applications-of-derivatives.md