# Product of a Derivative and its Inverse

1. Oct 10, 2011

### EconStudent

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. Oct 10, 2011

### Dick

Use that x/r=cos(theta). What does that make 1-(x/r)^2?

3. Oct 10, 2011

### EconStudent

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.

4. Oct 11, 2011

### 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
$$\frac{sin^2(x)}{1} -cos^2(x)$$