How do you differentiate #f(x)=cot(e^(1/x)) # using the chain rule?
Given function:
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To differentiate ( f(x) = \cot(e^{1/x}) ) using the chain rule, follow these steps:
- Recognize that the function ( f(x) ) is composed of two functions: the outer function ( \cot(x) ) and the inner function ( e^{1/x} ).
- Use the chain rule, which states that if ( y ) is a function of ( u ) and ( u ) is a function of ( x ), then ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ).
- Compute the derivative of the outer function ( \cot(x) ), which is ( -\csc^2(x) ).
- Compute the derivative of the inner function ( e^{1/x} ), which is ( -\frac{e^{1/x}}{x^2} ).
- Combine the derivatives using the chain rule: ( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = (-\csc^2(e^{1/x})) \cdot (-\frac{e^{1/x}}{x^2}) ).
- Simplify the expression if necessary.
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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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