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

The chain rule allows us to take the derivative of a composition of two or more functions. For two functions:

By the chain rule, we first need to take the derivative of the outermost function and work our way in. It's like taking the function apart layer by layer until we get all the way through.

This can also be written as:

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To differentiate ( f(x) = \tan(e^{1/x}) ) using the chain rule, follow these steps:

- Identify the outer function and the inner function. In this case, the outer function is ( \tan(x) ) and the inner function is ( e^{1/x} ).
- Compute the derivative of the outer function with respect to its argument. The derivative of ( \tan(x) ) is ( \sec^2(x) ).
- Compute the derivative of the inner function with respect to ( x ). The derivative of ( e^{1/x} ) can be found using the chain rule as follows: ( \frac{d}{dx}(e^{1/x}) = e^{1/x} \cdot (-1/x^2) ).
- Substitute the inner function into the derivative of the outer function.
- Multiply the results from steps 2 and 3 to get the derivative of the composite function ( f(x) ).

Therefore, the derivative of ( f(x) = \tan(e^{1/x}) ) is:

[ f'(x) = \sec^2(e^{1/x}) \cdot e^{1/x} \cdot (-1/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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