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Find the derivative of the function.

$ f(t) = \tan (\sec(\cos t)) $

$=-\sec ^{2}(\sec (\cos t)) \sec (\cos t) \tan (\cos t) \sin t$

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we're going to use the chain rule to find the derivative of this function. And here we have an inside layer co sign a middle layer, see camp and an outside lawyer tangent. So there's three derivatives who are multiplying together. So we start with the outermost layer tangent, and it's derivative is he can't squared. So we have c can't squared of C can't of co sign t. Now we move on to the second layer. See can't. And the derivative of C camp is C camp times tangent. So we have C can't of co sign t times tangent of co sign t. Then we move on to the inside layer, which is co sign and it's derivative is negative signs. We have negative sign of teeth now. There's really nothing that combines here. So the only thing we could do for simplifying is to take this negative sign and bring it out to the front. Kind of a shame to have to rewrite the whole thing just to do that. Um, but that's convention now. It doesn't matter what order we put each of these terms in, So if you look it up in the back of the book, and they have the terms in a different order. That's okay, because it's just a product, and we know that we could multiply things in a variety of orders. That's the community of property, of multiplication, and there we have our derivative.

Oregon State University