INVERSE TRIGONOMETRIC DERIVATIVE HELP

[NIPUL JARIWALA]

How to find the derivative of [cos^-1(x)]/x using first principle.

USING FIRST PRINCIPLE

PLEASE HELP

 

[Dr.Peterson]

Please show what you have tried, so we can see where you need help. I presume we don't need to teach you the definition of the derivative, so you must be stuck somewhere further along ...

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I have to admit I would never even try doing this if I didn't have to; that is why we have theorems like the quotient rule! But it's quite possible that the proof of the quotient rule could be instructive.

 

[HallsofIvy]

"Using first principles"? I would interpret that as meaning to use the limit, as h goes to 0 of \(\displaystyle \frac{\frac{cos^{-1}(x+ h)}{x+ h}- \frac{cos^{-1}(x)}{x}}{h}\).

Wow, I would hate to have to do that! Do you know any trig identity for \(\displaystyle cos^{-1}(a+ b)\)

 

[NIPUL JARIWALA]

I got the answer.
Is this method correct for first principle?

 

[Dr.Peterson]

How do you define "using first principles"? This looks like at least "second or third principles" to me!

 

[NIPUL JARIWALA]

This is an example from a book where i got the name of the method, "first principle".

 

[Dr.Peterson]

Interesting. You're effectively differentiating implicitly using the definition. Typically I've seen "first principles" used to mean directly applying the definition, not in this sort of more indirect usage. Now I know.

 

[Steven G]

I never remember the derivative of inverse trig function as I feel that they are a waste of time to memorize. I derive them whenever I need them and it does not take long at all. It actually looks similar to what you do.

y = sin^-1(x). so sin(y) = x. Taking the derivative of both sides gives me cos(y)y' = 1. So y' = 1/cos(y).

Now I can easily draw a triangle that obeys sin(y) = x = x/1 to get that cos(y) = sqrt(1-x²) and y' = 1/sqrt(1-x²)

 

[Steven G]

For you problem you have y = cos^-1(x)/x. So cos(xy) = x

Then -sin(xy)(y + xy') = 1. So -1/sin(xy) = y + xy' and y' = -(1/sin(xy) + y)/x

So what is sin(xy) in terms of just x? cos(xy) = x so sin(xy) = sqrt(1-x²) ....

 

[Steven G]

May I please ask what country the textbook is from?

 

[NIPUL JARIWALA]

I am from India.

I could have find the answer using CUOTIENT RULE. But I was asked to get the answer using FIRST PRINCIPLE. I am taught derivative from basic to advance. For that I have to practice all methods.

There are some other pages of the book