How do you differentiate #f(x) = arcsin(2x + 1)#?

Answer 1

The final answer is #1/sqrt(-x(x+1))#
This is found using the standard result for differentiating arcsine, and the chain rule.

Solution

#f(x)=arcsin(2x+1)# let #y=f(x)# (I just find it easier to explain in this notation.)
We know that #d/dx(arcsin(x))=1/sqrt(1-x^2)# (See below for derivation) apply chain rule so #dy/dx=dy/(du)*(du)/(dx)# and let #u=2x+1# so #y=arcsin(u)# #dy/(du)=1/sqrt(1-u^2)#
#u=2x+1# #(du)/(dx)=2#
So #dy/dx=(1/sqrt(1-u^2))*2#
#u^2=4x^2 +4x +1#
#dy/dx=2/sqrt(1-4x^2-4x-1)#
#dy/dx=2/sqrt(-4x(x+1))#
#dy/dx=2/(2sqrt(-x(x+1))#
#dy/dx=1/sqrt(-x(x+1))#
Standard result for the derivative of arcsine function derivation. let #y=arcsin(x)# therefore #x=siny# differentiate implicitly with respect to x #1=cosydy/dx# rearrange #dy/dx=1/cosy# use fundamental trig identity: #(sinx)^2 + (cosx)^2 =1# so #cosy=sqrt(1-(siny)^2)# but #x=siny# so #cosy=sqrt(1-x^2)# therefore #dy/dx=1/sqrt(1-x^2)#
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Answer 2

To differentiate ( f(x) = \arcsin(2x + 1) ), we apply the chain rule. The derivative of ( \arcsin(u) ) is ( \frac{1}{\sqrt{1 - u^2}} ), and then we multiply by the derivative of the inner function. In this case, the inner function is ( 2x + 1 ). So, the derivative of ( f(x) ) with respect to ( x ) is:

[ f'(x) = \frac{1}{\sqrt{1 - (2x + 1)^2}} \cdot (2) ]

Simplifying this expression gives the derivative of ( f(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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