Parametric & Implicit Differentiation

Master advanced derivative techniques for parametric curves and implicit relations. Learn to calculate tangent lines, interpret rates of change, and apply these concepts to real-world modeling scenarios.

12th Grade

Calculus

~90 min

🎮 Interactive Activity: Parametric Derivative Calculator

Calculate dy/dx for the parametric curve!

Parametric Equations:

x(t) = 2t

y(t) = 3t²

At t = 1

dy/dx = ?

🎮 Interactive Activity: Implicit Differentiation Practice

Find dy/dx for the implicit equation!

Implicit Equation:

x² + y² = 13

At point (2, 3)

dy/dx = ?

1. Parametric Differentiation

Understanding Parametric Curves

Parametric equations define curves using a parameter t. Instead of y = f(x), we have x = f(t) and y = g(t). This allows us to describe more complex curves, including those that aren't functions.

Example 1: Basic Parametric Derivative

Given: x = 2t, y = 3t²

Step 1: Find dx/dt = 2

Step 2: Find dy/dt = 6t

Step 3: Apply chain rule: dy/dx = (dy/dt)/(dx/dt) = 6t/2 = 3t

Answer: dy/dx = 3t

Example 2: Trigonometric Parametric Curve

Given: x = cos(t), y = sin(t) (a circle)

Step 1: dx/dt = -sin(t), dy/dt = cos(t)

Step 2: dy/dx = cos(t)/(-sin(t)) = -cot(t)

At t = π/4: dy/dx = -cot(π/4) = -1

Answer: dy/dx = -cot(t)

Example 3: Second Derivative

Given: x = t², y = t³

First derivative: dy/dx = (3t²)/(2t) = 3t/2

Second derivative: d²y/dx² = d/dx(dy/dx) = d/dt(3t/2) / (dx/dt) = (3/2)/(2t) = 3/(4t)

Answer: d²y/dx² = 3/(4t)

2. Implicit Differentiation

3. Chain Rule in Parametric & Implicit Forms

4. Tangent Lines and Normal Lines

5. Real-World Applications

6. Advanced Techniques

7. Optimization with Parametric and Implicit Curves

Frequently Asked Questions

Practice Time!

Practice Quiz

Questions

Correct

Score

For the parametric curve x = 2t, y = 3t², what is dy/dx at t = 1?

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For the implicit equation x² + y² = 25, what is dy/dx at the point (3, 4)?

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If x = cos(t) and y = sin(t), what is dy/dx?

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For x²y + y² = 10, find dy/dx using implicit differentiation.

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A particle moves along x = t², y = t³. What is the slope of the tangent at t = 2?

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For the implicit equation xy = 4, what is dy/dx?

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If x = e^t and y = e^(2t), what is dy/dx?

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For x² + xy + y² = 7, find dy/dx at (1, 2).

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A curve is given by x = 3t, y = 4t². What is the equation of the tangent line at t = 1?

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For the implicit equation y² = x³, what is dy/dx?

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