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Let's now use what we know about the Chain rule and the Product rule

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To take the derivative of an even weirder expressions.

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We are going to take the derivative of e^cos(x) times cos(e^x)

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Let's take the derivative of this.

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So we can view this as the product of two functions,

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so the product rule tells us that this is going to be the derivative with the respect to x

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of e^cos(x) times cos(e^x) plus the first function

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just e^cos(x) times the derivative of the second function cos(e^x)

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And so we just need to figure out what these two derivatives are.

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So we can use the Chain Rule here.

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Just to make it clear, we got this from the Product Rule.

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But then to evaluate each of these derivatives, we need to use the Chain Rule.

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So let's think about this a little bit.

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So let's think about what the derivative of this function is.