How do you differentiate #x * (4-x^2)^(1/2)#?

Answer 1

I found: #2(2-x^2)/(sqrt(4-x^2))#

You can use the Product Rule to deal with the product between the two functions and the Chain Rule to deal with the #()^(1/2)#: #y'=1*(4-x^2)^(1/2)+1/2x(4-x^2)^(1/2-1)*(-2x)=# #=(4-x^2)^(1/2)-x^2(4-x^2)^(-1/2)=# Considering that: #(4-x^2)^(1/2)=sqrt(4-x^2)# and #(4-x^2)^(-1/2)=1/sqrt(4-x^2)# You get: #sqrt(4-x^2)-x^2/(sqrt(4-x^2))=(4-x^2-x^2)/(sqrt(4-x^2))=# #=(4-2x^2)/(sqrt(4-x^2))=2(2-x^2)/(sqrt(4-x^2))#
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Answer 2

To differentiate ( x \cdot (4-x^2)^{\frac{1}{2}} ), use the product rule and the chain rule.

  1. Apply the product rule: ((uv)' = u'v + uv').
  2. Let ( u = x ) and ( v = (4-x^2)^{\frac{1}{2}} ).
  3. Find ( u' ) and ( v' ) using the chain rule.
    • ( u' = 1 )
    • ( v' = \frac{1}{2}(4-x^2)^{-\frac{1}{2}} \cdot (-2x) )
  4. Plug the values into the product rule formula: [ \frac{d}{dx}(x \cdot (4-x^2)^{\frac{1}{2}}) = 1 \cdot (4-x^2)^{\frac{1}{2}} + x \cdot \frac{1}{2}(4-x^2)^{-\frac{1}{2}} \cdot (-2x) ]
  5. Simplify the expression.
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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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