# What is the second derivative of #f(x)=cos(x^2) #?

We use chain rule here

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Here,

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To find the second derivative of ( f(x) = \cos(x^2) ), we first find the first derivative using the chain rule, and then differentiate again using the chain rule and product rule.

The first derivative is: [ f'(x) = -2x \sin(x^2) ]

To find the second derivative, we use the product rule and chain rule: [ f''(x) = \frac{d}{dx} (-2x) \sin(x^2) + (-2x) \frac{d}{dx} \sin(x^2) ]

Simplifying and applying the chain rule for the derivative of ( \sin(x^2) ): [ f''(x) = -2 \sin(x^2) - 4x^2 \cos(x^2) ]

So, the second derivative of ( f(x) = \cos(x^2) ) is ( f''(x) = -2 \sin(x^2) - 4x^2 \cos(x^2) ).

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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.

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