Purpose of the derivative of the inverse function

[barryj]

Homework Statement

In calculus, I learn that the derivative of the inverse function is

g'(x) = 1/ f'(g(x))

Homework Equations

So..

The Attempt at a Solution

Can someone give me an example of where I need to know this, or is this just a math exercise. Is there a relatively simple physics example?

 

[FactChecker]

Sure. Suppose you are controlling an engine speed with a dial. If you want to increase the speed by 3 rpm, how much should you increase the dial setting?

PS. Keep in mind that this gives a linear estimate at that setting. It maybe very different at another setting.

 

[barryj]

Don't go away for long. I will be back in4 hours.

 

[FactChecker]

This property of the derivative of the inverse function is just a simple observation of what a slope is when looked at from another direction. As such, it has applications everywhere.

 

[member 587159]

What is the derivative of $\arccos x$? Good luck with using the definition.

 

[barryj]

OK, finding the derivative of arccos(x)
given... f(x) = cos(X) and g(x) = cos^-1(x)

then g'(x) = 1/f'(g(x))

g'(x) = 1/-sin(g(x))

g'(x) = 1/-sin(cos^-1(x))

then after some trig substitutions we get this is equal to $-1/\sqrt(1-u^2)$or something like this.

 

[member 587159]

OK, finding the derivative of arccos(x)
given... f(x) = cos(X) and g(x) = cos^-1(x)

then g'(x) = 1/f'(g(x))

g'(x) = 1/-sin(g(x))

g'(x) = 1/-sin(cos^-1(x))

then after some trig substitutions we get this is equal to $-1/\sqrt(1-u^2)$or something like this.

This is one of the numerous examples where it is useful.

 

[barryj]

Could someone give me a few more examples where finding the derivative of the inverse function is useful.