How do you differentiate #x * (4-x^2)^(1/2)#?
I found:
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To differentiate ( x \cdot (4-x^2)^{\frac{1}{2}} ), use the product rule and the chain rule.
- Apply the product rule: ((uv)' = u'v + uv').
- Let ( u = x ) and ( v = (4-x^2)^{\frac{1}{2}} ).
- Find ( u' ) and ( v' ) using the chain rule.
- ( u' = 1 )
- ( v' = \frac{1}{2}(4-x^2)^{-\frac{1}{2}} \cdot (-2x) )
- 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) ]
- Simplify the expression.
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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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