How do you differentiate #f(x)=cos(sqrt((cosx^2))) # using the chain rule?
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I'll be using Leibniz notation because I think it's easier to understand more complicated chain rule applications with this.
The chain rule, in said notation goes as follows,
In this case we have
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To differentiate ( f(x) = \cos(\sqrt{\cos(x^2)}) ) using the chain rule, follow these steps:
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Identify the outer function and inner function.
- Outer function: ( \cos(x) )
- Inner function: ( \sqrt{\cos(x^2)} )
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Find the derivative of the outer function with respect to its inner function.
- ( \frac{d}{du} \cos(u) = -\sin(u) ), where ( u = \sqrt{\cos(x^2)} )
-
Find the derivative of the inner function with respect to ( x ).
- ( \frac{d}{dx} \sqrt{\cos(x^2)} = \frac{-\sin(x^2)}{2\sqrt{\cos(x^2)}} )
-
Apply the chain rule:
- ( \frac{d}{dx} \cos(\sqrt{\cos(x^2)}) = -\sin(\sqrt{\cos(x^2)}) \times \frac{-\sin(x^2)}{2\sqrt{\cos(x^2)}} )
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Simplify the expression if needed.
So, the derivative of ( f(x) = \cos(\sqrt{\cos(x^2)}) ) using the chain rule is: [ -\sin(\sqrt{\cos(x^2)}) \times \frac{-\sin(x^2)}{2\sqrt{\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.
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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