How do you differentiate #f(x)=cot(sqrt(x^21)) # using the chain rule?
The explanation is given below.
Let us first write the same as the following.
Chain rule would work like this:
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To differentiate the function (f(x) = \cot(\sqrt{x^21})) using the chain rule, follow these steps:

Recognize the composition of functions: The function (f(x) = \cot(\sqrt{x^21})) is a composition of several functions:
 (u(x) = x^2  1)
 (v(u) = \sqrt{u})
 (w(v) = \cot(v)) So, (f(x) = w(v(u(x)))).

Apply the chain rule: The chain rule states that if a function (y = f(g(x))), then its derivative (y') can be found as (f'(g(x)) \cdot g'(x)). For our composite function, we'll apply the chain rule multiple times.

Find the derivatives of the inner functions:
 For (u(x) = x^2  1), the derivative (u'(x) = 2x).
 For (v(u) = \sqrt{u}), applying the derivative of a square root function (v'(u) = \frac{1}{2\sqrt{u}}).
 For (w(v) = \cot(v)), the derivative (w'(v) = \csc^2(v)).

Apply the derivatives according to the chain rule: [f'(x) = w'(v(u(x))) \cdot v'(u(x)) \cdot u'(x)]
Substitute the derivatives we found: [f'(x) = \csc^2(\sqrt{x^21}) \cdot \frac{1}{2\sqrt{x^21}} \cdot 2x]
Simplify: [f'(x) = \frac{2x}{2\sqrt{x^21}} \cdot \csc^2(\sqrt{x^21})] [f'(x) = \frac{x \cdot \csc^2(\sqrt{x^21})}{\sqrt{x^21}}]
So, the derivative of (f(x) = \cot(\sqrt{x^21})) with respect to (x) is (f'(x) = \frac{x \cdot \csc^2(\sqrt{x^21})}{\sqrt{x^21}}).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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