# How do you differentiate #y = arccosx + x sqrt(1-x^2)#?

What you want to do is break the problem down into parts.

#1. First you want to find the derivative of arccosx...

#2. Next you want to find the derivative of x*(1-x^2)^(1/2)...

#3. Once you have these two derivatives, you can now combine them to produce your final result (dy/dx)...

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To differentiate ( y = \arccos(x) + x\sqrt{1-x^2} ), you can use the sum rule and the chain rule of differentiation. The derivative of ( \arccos(x) ) with respect to ( x ) is ( -\frac{1}{\sqrt{1-x^2}} ), and the derivative of ( x\sqrt{1-x^2} ) can be found using the product rule. The derivative of ( \sqrt{1-x^2} ) is ( -\frac{x}{\sqrt{1-x^2}} ).

Therefore, the derivative of ( y ) with respect to ( x ) is:

[ \frac{dy}{dx} = -\frac{1}{\sqrt{1-x^2}} + \sqrt{1-x^2} - \frac{x^2}{\sqrt{1-x^2}} ]

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