# How do you use the chain rule to differentiate #y=e^(sinx)#?

The chain rule states that:

and in conclusion:

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To differentiate ( y = e^{\sin(x)} ) using the chain rule:

- Recognize that ( y ) is a composition of two functions: the outer function is ( e^u ) and the inner function is ( \sin(x) ).
- Apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
- Derivative of the outer function ( e^u ) with respect to ( u ) is ( e^u ).
- Derivative of the inner function ( \sin(x) ) with respect to ( x ) is ( \cos(x) ).
- Combine these derivatives to get the derivative of ( y = e^{\sin(x)} ) with respect to ( x ): [ \frac{dy}{dx} = e^{\sin(x)} \cdot \cos(x) ] or, equivalently, [ \frac{dy}{dx} = \cos(x) \cdot e^{\sin(x)} ]

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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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