# How do you differentiate #f(x)=e^cot(1/sqrt(x)) # using the chain rule?

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To differentiate ( f(x) = e^{\cot\left(\frac{1}{\sqrt{x}}\right)} ) using the chain rule, follow these steps:

- Identify the outer function as ( e^u ), where ( u = \cot\left(\frac{1}{\sqrt{x}}\right) ).
- Differentiate the outer function with respect to its inner variable ( u ): ( \frac{d}{du}(e^u) = e^u ).
- Identify the inner function as ( \cot\left(\frac{1}{\sqrt{x}}\right) ).
- Differentiate the inner function with respect to its variable.
- Let ( v = \frac{1}{\sqrt{x}} ).
- Differentiate ( v ) with respect to ( x ): ( \frac{dv}{dx} = -\frac{1}{2x^{3/2}} ).
- Use the chain rule to differentiate ( \cot(v) ) with respect to ( x ): ( \frac{d}{dx}(\cot(v)) = \frac{d}{dv}(\cot(v)) \cdot \frac{dv}{dx} ).

- Find the derivative of ( \cot(v) ) with respect to ( v ): ( \frac{d}{dv}(\cot(v)) = -\csc^2(v) ).
- Substitute ( v = \frac{1}{\sqrt{x}} ) into ( \cot(v) ) and ( -\frac{1}{2x^{3/2}} ) into ( \frac{dv}{dx} ).
- Multiply the derivatives obtained in steps 4 and 5 to get the derivative of the inner function with respect to ( x ).
- Multiply the derivative obtained in step 7 by the derivative of the outer function obtained in step 2 to find the derivative of the entire function ( f(x) ) with respect to ( x ).

Combining all these steps, the derivative of ( f(x) ) using the chain rule is:

[ f'(x) = e^{\cot\left(\frac{1}{\sqrt{x}}\right)} \cdot (-\csc^2\left(\frac{1}{\sqrt{x}}\right)) \cdot \left(-\frac{1}{2x^{3/2}}\right) ]

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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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