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Polynomial Rule for Derivatives: Exponential Functions and Higher Order Derivatives

This review covers the polynomial rule for derivatives, differentiating exponential functions, higher order derivatives, and how to differentiate combinations of functions including the product and quotient rules. It also explores differentiating and integrating polynomials and the slope of exponential functions. Additionally, the chain rule and functions of functions are discussed.

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Polynomial Rule for Derivatives: Exponential Functions and Higher Order Derivatives

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  1. Winter wk 4 – Tues.25.Jan.05 • Review: • Polynomial rule for derivatives • Differentiating exponential functions • Higher order derivatives • How to differentiate combinations of functions? • Product rule (3.3) • Quotient rule (3.4) Energy Systems, EJZ

  2. Differentiating polynomials and ex Differentiating polynomials: Integrating polynomials: Slope of ex increases exponentially: d/dx(ex) = ex d/dx(ax) = ln(a) ax

  3. Higher order derivatives Second derivative = rate of change of first derivative

  4. Ch.3.3: Products of functions If these are plots of f(x) and g(x) Then sketch the product y(x) = f(x).g(x) = f.g

  5. Differentiating products of functions Ex: We couldn’t do all derivatives with last week’s rules: y(x) = x ex. What is dy/dx? Write y(x) = f(x) g(x). Slope of y = (f * slope of g) + (g * slope of f) Try this for y(x) = x ex, where f=x, g=ex

  6. Proof (justification)

  7. Practice – Ch.3.3 Spend 10-15 minutes doing odd # problems on p.121 Pick one or two of these to set up together: 32, 38, 45

  8. Ch.3.4 Functions of functions Ex: We couldn’t do 3.1 #36 with last week’s rules: y =(x+3)½ What is dy/dx? Consider y(x) = f(g(x)) = f(z), where z=g(x). Try this for y =(x+3)½ , where z=x+3, f=z½

  9. Proof (justification) Differentiating functions of functions: y(x) = f(g(x)) See #17, p.154 Derive the chain rule using local linearizations: g(x+h) ~ g(x) + g’(x) h = f(z+k) ~ f(z) + f’(z) k = y’ = f’(g(x)) =

  10. Differentiating functions of Functions If these are plots of f(x) and g(x) Then sketch function y(x) = f(g(x)) = f(g)

  11. Candidates for y= f(g(x))

  12. Answer

  13. Calc Ch.3.4 Conceptest 2

  14. Calc Ch.3.4 Conceptest 2 options

  15. Calc Ch.3.4 Conceptest 2 answer

  16. Calc Ch.3.4 Conceptest 3

  17. Calc Ch.3.4 Conceptest 3 options

  18. Calc Ch.3.4 Conceptest 3 answer

  19. Practice – Ch.3.4 Spend 10-15 minutes doing odd # problems on p.126 Pick one or two of these to set up together: 52, 54, 62, 66, 68

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