Question

if y=sin^-1(5x), then dy/dx=

Answer 1

is it \[\sin^{-1}(5x)\] ?

Answer 2

yea!

Answer 3

is that the answer you came up with?

Answer 4

okay, so you understand that this is arcsin(5x) right ?

if you know the formula, you can use that.

what I recommend though, is implicit differentiation.
are you familiar with it ?

Answer 5

Erm...I have a pretty concise way of demonstrating the derivation of the derivative of arcsin, let me know if you'd like to see it ayanamarcell. :)

Answer 6

implicit differeniation?..no im not familar with it.

Answer 7

you go like this

y = arcsin(5x) so using inverse relations,

sin(y) = 5x

when you differentiate both sides, do you know what happens?

Answer 8

Yuki, there's a better way! :D (Well, if he/she's not familiar with implicit differentiation that is)

Answer 9

@Yuki: no not really

Answer 10

Sorry Yuki, you were going there. :P But technically, you don't need implicit differentiation for this type of problem (check my file for clarification).

Answer 11

okay, I don't want to force you down another thing you need
to learn, so let's see what QM tells us.

I am very interested

Answer 12

@QM: I cant open your doc

Answer 13

Ahh. :/ Darn. Then let me put my derivation here:

Answer 14

thanks Yuki

Answer 15

\[y = \sin^{-1} x\]\[x = \sin(y)\]\[\frac{dx}{dy} = \cos(y)\] (invert to get dy/dx) \[\frac{dy}{dx} = \frac{1}{\cos(y)}\] Using the main pythagorean identity,\[\frac{dy}{dx} = \frac{1}{\sqrt{1 - \sin^2(y)}}\] And because sin^2(y) = x^2 from our initial conditions,\[\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\]

Answer 16

Because you only have a 5x inside the operator instead of x, you can multiply everything by the derivative of 5x (chain rule) and you have your derivative!

Answer 17

Wow that was a GOOD one !

Answer 18

(And, of course, the x^2 inside the square root will be a (5x)^2, to keep things consistent.) Thanks yuki!

Answer 19

ayanamarcell, the one with implicit differentiation will be
very similar to this proof, so if you understand this one
you should not have problem learning it.

Answer 20

thanks for the help! that was really simple to understand.

Answer 21

wat about sin^-1 sqr x