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DRILL

DRILL. Find the derivative of the following: y = sin 2 (3x – 2) y = cos 2 (7x). y = sin 2 (3x – 2). Let u = 3x – 2; du/ dx = 3 y = sin 2 u y = sin u •sin u Using the chain rule, dy /du = sin (u) cos (u) + sin (u) cos (u) = 2sin(u) cos (u)

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DRILL

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  1. DRILL • Find the derivative of the following: • y = sin2 (3x – 2) • y = cos2 (7x)

  2. y = sin2 (3x – 2) • Let u = 3x – 2; du/dx = 3 • y = sin2 u • y = sin u •sin u • Using the chain rule, dy/du = sin (u) cos (u) + sin (u) cos(u) = 2sin(u)cos(u) • From trig identities: 2sin(u)cos(u) = sin 2u  sin 2(3x -2) = sin (6x – 4) • (du/dx)(dy/du) = 3sin(6x -4)

  3. y = cos2 (7x) • Let u = 7x; du/dx = 7 • y = cos2u • dy/du = cosu(-sinu) + cosu(-sinu) = -2cos(u)sin(u) = -sin2u = -sin2(7x) • (du/dx)(dy/dx) = 7 (-sin2(7x)) =-7sin(14x)

  4. Implicit Differentiation Lesson 3.7

  5. Objectives • Students will be able to • find derivatives using implicit differentiation. • find derivatives using the Power Rule for Rational Powers of x.

  6. Find dy/dx if y2 = x • To find dy/dx, we simply differentiate both sides of the equation with respect to x, treating y as a differentiable function of x and applying the chain rule. • y2 = x • 2y dy/dx = 1 • dy/dx = 1/(2y)

  7. Find

  8. More Examples

  9. More Examples

  10. More Examples

  11. Homework • P. 162: 2-14 • THIS IS A CHANGE FROM THE ORIGINAL • On 9-12, do implicit differentiation, then substitute the given values for x and y into your answer. • On 13-16, do implicit differentiation, then using dy/dx equation, determine where the curve would be defined (like when the denominator ≠ 0)

  12. Example: p. 162: #17 • Find the lines that are tangent and normal to x2 + xy – y2 = 1 at (2, 3) • 2x + x•1 (dy/dx) + y • 1 – 2y (dy/dx) = 0 • 2x + x(dy/dx) + y – 2y(dy/dx) = 0 • x(dy/dx) – 2y(dy/dx) = -2x – y • dy/dx = (-2x – y)/(x-2y) • x = 2, y = 3, slope (tangent) = -7/-4 = 7/4

  13. Example: p. 162: #17 • Equation of tangent: m = 7/4 through (2, 3) • y – 3 = 7/4 (x – 2) • y – 3 = (7/4)x – 7/2 • y = (7/4)x – ½ • Equation of normal: m = -4/7 through (2, 3) • y – 3 = -4/7 (x – 2) • y – 3 = (-4/7)x +8/7 • y = (-4/7) x + 29/7

  14. Finding a Second Derivative Implicitly

  15. Finding a Second Derivative Implicitly

  16. Finding a Second Derivative Implicitly

  17. Finding a Second Derivative Implicitly

  18. Finding a Second Derivative Implicitly

  19. More Examples

  20. More Examples

  21. More Examples

  22. Homework • Page 162: 20-26 EVEN, 27-39 (odd) • THIS IS A CHANGE FROM THE ORIGINAL

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