Chapter 8

Differentiation

Chapter description
In this chapter we look at how it is possible to describe relationships which are changing and how we can find out the rate of this change.
The chapter is split up into the following sections.

  1. Some problems answered and difficulties solved
    1. How can we find a speed from knowing the distance travelled?
    2. How does y = xn change as x changes?
    3. Different ways of writing differentiation
    4. Some special cases of y = axn
    5. Differentiating x = cos t answers another thinking point
    6. Can we always differentiate? If not, why not?

  2. Natural growth and decay -- the number e
    1. Even more money -- compound interest and exponential growth
    2. What is the equation of this smooth growth curve?
    3. Getting numerical results from the natural growth law of x = et
    4. Relating ln x to the log of x using other bases
    5. What do we get if we differentiate ln t?

  3. Differentiating more complicated functions
    1. The Chain Rule
    2. Writing the Chain Rule in function form
    3. Differentiating functions with angles in degrees or logs to base 10
    4. The Product Rule, or 'uv' Rule
    5. The Quotient Rule, or 'u/v' Rule

  4. The hyperbolic functions of sinh x and cosh x
    1. Getting symmetries from ex and e-x
    2. Differentiating sinh x and cosh x
    3. Using sinh x and cosh x to get other hyperbolic functions
    4. Comparing other hyperbolic and trig formulas -- Osborn's Rule
    5. Finding the inverse function for sinh x
    6. Can we find an inverse function for cosh x?
    7. tanh x and its inverse function
    8. What's in a name? Why 'hyperbolic' functions?
    9. Differentiating inverse trig and hyperbolic functions

  5. Some uses for differentiation
    1. Finding the equations of tangents to particular curves
    2. Finding turning points and points of inflection
    3. General rules for sketching curves
    4. Some practical uses of turning points
    5. A clever use for tangents -- the Newton-Raphson Rule

  6. Implicit differentiation
    1. How it works, using circles as examples
    2. Using implicit differentiation with more complicated relationships
    3. Differentiating inverse functions implicitly
    4. Differentiating exponential functions like x = 2t
    5. A practical application of implicit differentiation

  7. Writing functions in an alternative form using series
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