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What is the derivative of #y=arcsin(x/3 )#?

Answer 1

Luke Alvarez

First recall the definition for derivative of #arcsinx#:

#d/dx[arcsinx] = 1/sqrt(1-x^2)#.

Since we're differentiating with #x/3# instead of #x#, we need to substitute and apply the chain rule:

#d/dx[arcsin(x/3)] = d/dx[x/3] * 1/(sqrt(1-(x/3)^2))#

Simplifying yields:

#d/dx[arcsin(x/3)] = 1/(3sqrt(1-x^2/9)) = 1/(sqrt(9-x^2))#

A page explaining this simplification in more detail can be found here .

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Answer 2

Adam Adkins

The derivative of ( y = \arcsin(\frac{x}{3}) ) with respect to ( x ) is:

[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - \left(\frac{x}{3}\right)^2}} \times \frac{1}{3} = \frac{1}{3\sqrt{1 - \left(\frac{x}{3}\right)^2}} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.