What is the derivative of this function #f(x) = sin (1/x^2)#?
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The derivative of the function ( f(x) = \sin(1/x^2) ) is:
[ f'(x) = -\frac{2\cos(1/x^2)}{x^3} ]
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To find the derivative of the function ( f(x) = \sin\left(\frac{1}{x^2}\right) ), we can use the chain rule.
Let ( u = \frac{1}{x^2} ), then ( \frac{du}{dx} = -\frac{2}{x^3} ).
Now, applying the chain rule, we have:
[ \frac{df}{dx} = \cos\left(\frac{1}{x^2}\right) \cdot \frac{d}{dx}\left(\frac{1}{x^2}\right) ]
[ = \cos\left(\frac{1}{x^2}\right) \cdot \left(-\frac{2}{x^3}\right) ]
[ = -\frac{2 \cos\left(\frac{1}{x^2}\right)}{x^3} ]
Therefore, the derivative of the function ( f(x) = \sin\left(\frac{1}{x^2}\right) ) is ( -\frac{2 \cos\left(\frac{1}{x^2}\right)}{x^3} ).
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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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