How do you find the derivative of #y=tan(arcsin(x))# ?

Answer 1
The answer is #sec^2(arcsin(x))*1/sqrt(1-x^2)#.
You have to recognize that the function is a composition of functions. Then you will understand that you will apply the chain rule: #(dy)/(dx)=f'(g(x))*g'(x)#.
So, let #f(x)=tan(x)# and #g(x)=arcsin(x)#. Then #f'(x)=sec^2(x)# and #g'(x)=1/sqrt(1-x^2)#.

Substitutions into the chain rule should then be implemented.

Writing a reference sheet with all of your basic derivatives and practicing the chain rule is the most effective way to study this material. Since integration heavily relies on this concept, it is an essential subject to learn!

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Answer 2

To find the derivative of ( y = \tan(\arcsin(x)) ), you can use the chain rule of differentiation. The chain rule states that if you have a function ( f(g(x)) ), then its derivative is given by ( f'(g(x)) \cdot g'(x) ).

Let's break down the given function ( y = \tan(\arcsin(x)) ):

  • Let ( u = \arcsin(x) ), so ( y = \tan(u) ).
  • Find the derivative of ( u ) with respect to ( x ), which is ( \frac{du}{dx} = \frac{1}{\sqrt{1-x^2}} ) using the derivative of arcsine.
  • Now, find the derivative of ( y = \tan(u) ) with respect to ( u ), which is ( \sec^2(u) ).

Apply the chain rule by multiplying these derivatives: [ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \sec^2(u) \cdot \frac{1}{\sqrt{1-x^2}} ]

Replace ( u ) with ( \arcsin(x) ) to get the final derivative: [ \frac{dy}{dx} = \sec^2(\arcsin(x)) \cdot \frac{1}{\sqrt{1-x^2}} ]

This derivative formula represents the rate of change of ( y ) with respect to ( x ) for the given function ( y = \tan(\arcsin(x)) ).

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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.

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.

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.

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.

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