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MTH 251 – Differential Calculus Chapter 3 – Differentiation

MTH 251 – Differential Calculus Chapter 3 – Differentiation. Section 3.6 Implicit Differentiation. Equations of Curves. Explicit : y = f(x) Set of ordered pairs (x, y) = (x, f(x)) 2 nd coordinate is given in terms of an expression involving the 1 st coordinate.

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MTH 251 – Differential Calculus Chapter 3 – Differentiation

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  1. MTH 251 – Differential CalculusChapter 3 – Differentiation Section 3.6 Implicit Differentiation

  2. Equations of Curves • Explicit: y = f(x) • Set of ordered pairs (x, y) = (x, f(x)) • 2nd coordinate is given in terms of an expression involving the 1st coordinate. • Parametric: x = f(t), y = g(t) • Set of ordered pairs (x, y) = (f(t), g(t)) • 1st coordinate is given in terms of an expression of the parameter, t • 2nd coordinate is given in terms of an expression of the parameter, t • Implicit: f(x,y) = 0 • Set of ordered pairs (x,y) such that f(x,y) = 0 • f(x,y) is an expression involving x and/or y

  3. Equations of Curves - Example • A Circle of Radius 2 • Explicit: • Parametric: • Implicit: • Line with slope 2/3 containing the point (0, 5) • Explicit: • Parametric: • Implicit:

  4. Explicit  Parametric AKA: Parameterization • If y = f(x) … • Let x = g(t) … any expression of t • Substitute to get y = f(x) = f(g(t)) • Determine the domain for t. • Example …

  5. Parametric Explicit AKA: Eliminating the Parameter • If x = f(t), y = g(t) … • Solve x = f(t) for t, giving t = h(x) • Substitute to get y = g(t) = g(h(x)) • Determine the domain for x. • Example …

  6. Explicit  Implicit • If y = f(x) … • Move everything to one side of the equation. • (optional) Simplify. • I.E. f(x) – y = 0 or y – f(x) = 0 • Example …

  7. Implicit  Explicit • If f(x, y) = 0 … • Solve for y ... if possible! • Examples …

  8. Implicit Differentiation • Finding dy/dx for an implicitly defined function without explicitly solving for y. • Note: The result may (will) be in terms of x & y • Differentiate both sides of the equation in terms of x, treating y as a function of x • i.e. use the chain rule and • Algebraically solve for dy/dx

  9. Implicit Differentiation - Examples

  10. Tangents & Normals • Tangent Line • The limit of secant lines. • Slope = dy/dx • Normal Line • The line perpendicular to the tangent. • Slope = –1/(dy/dx) • Example … find the tangent and normal to the curve y2 – 2x – 4y – 1 = 0 at the point (–2, 1)

  11. Second Derivatives Implicitly • Find the first derivative implicitly. • Differentiate the first derivative implicitly. • The answer will be in terms of dy/dx. • Substitute the 1st derivative into the 2nd derivative to get the result in terms of x and y only. • Higher Order Derivatives … continue likewise! • Example: Find the 1st & 2nd derivatives of …

  12. Power Rule … one more time What if n is a fraction?

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