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Derivative implicit functions

Implicit differentiation PowerPoint lesson that contrasts explicit and implicit functions, sets out the steps, and works through examples such as x² + y² = 4 and x³ + y³ − 9xy = 0.
Derivative implicit functions preview
Derivative implicit functions preview

Look Inside · 10 pages

Opening slide for the implicit differentiation lesson, stating the objective: differentiate implicit functions
Lesson title and learning objective
Review slide comparing explicit and implicit functions through the equivalent equations y = x² + 1 and y − x² = 1
Review of explicit and implicit functions
Implicit functions slide asking which of five equations are difficult to express explicitly, with marked choices
Five equations introducing implicit differentiation
Worked example rewriting y² = 4x as y = ±2√x and differentiating both branches to obtain dy/dx = 2/y
Example 1: differentiation of explicit branches
Method slide listing four steps for implicit differentiation: differentiate, collect dy/dx terms, factor, and solve
Four-step implicit differentiation method
Worked example differentiating y² = 4x implicitly with the chain rule, giving 2y(dy/dx) = 4 and dy/dx = 2/y
Example 1 revisited using implicit differentiation
Worked example differentiating x² + y² = 4 implicitly, using the chain rule and solving for dy/dx = −x/y
Example 2: implicit differentiation of a circle
Worked example differentiating x² + y³ = 8 implicitly, using the chain rule to obtain dy/dx = −2x/(3y²)
Example 3: differentiating an equation with y³
Worked example using the product rule on x + x²y + y³ = 100, then collecting and factoring dy/dx terms
Example 4: applying the product rule to x²y
Worked example differentiating x³ + y³ − 9xy = 0 with the product rule, yielding dy/dx = (3y − x²)/(y² − 3x)
Example 5: product rule and derivative simplification
File type
Microsoft PowerPoint
Use
Digital activity

Updated Jan 02, 2023

Subject
Maths
Topic
Calculus, Diffrentiation
Resource types
Lesson Plans
Use
Digital activity

From the author

About this resource

PowerPoint presentation, 15 slides explaining how to differentiate implicit functions

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In the classroom

Teaching tips & learning objectives

Learning objectives

  • Differentiate implicitly defined equations with respect to x

  • Apply the product rule within implicit differentiation

  • Find the gradient of a tangent and the equation of a normal line on an implicit curve

Skills covered

  • Step-by-step differentiation — following worked solutions for equations like x³ sin y + y = 4x + 3

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Resource details

Concepts & topics

Core concepts

  • Implicit differentiation

  • Product rule

  • Tangent line

Topics

Calculus Diffrentiation Implicit Function

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Categories this resource belongs to

Maths Lesson Plans Teacher Tools Year 13 Years 10–13

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