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Product of a Derivative and its Inverse

  1. Oct 10, 2011 #1
    1. The problem statement, all variables and given/known data

    For the functions:

    x = r*cos(θ)
    y = r*sin(θ)

    Calculate dθ/dX * dX/dθ (considering θ as a function of x, y) and simplify.

    3. The attempt at a solution

    I believe this should always be 1, by definition (as the product of a derivative and its inverse). However, I don't know how to show this.

    Solving the first function for θ, I get cos-1(x/r). The derivative of this is a relatively messy expression 1 over the product of r and a square root, and its being multiplied by something relatively simple (-r*sin(θ)). Why is it that this would simplify to 1?

    I get that the negatives and the r's will cancel. So it simplifies to:

    sin(θ) / √(1 - x2/r2)

    But I can't get any further nor am I satisfied that this expression is or is not equal to 1. Thanks in advance.
     
  2. jcsd
  3. Oct 10, 2011 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Use that x/r=cos(theta). What does that make 1-(x/r)^2?
     
  4. Oct 10, 2011 #3
    That's so clever, thank you. Substituting for x and squaring the top and bottom, we get sin2/1-cos2 which is just sin2/sin2.
     
  5. Oct 11, 2011 #4

    Mark44

    Staff: Mentor

    When you write fractional expressions in a line, and there are multiple terms in the top or bottom, USE PARENTHESES!

    What you wrote would be reasonably interpreted as
    [tex]\frac{sin^2(x)}{1} -cos^2(x)[/tex]
     
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