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Find the derivative of the function.$ f(t) = \tan (\sec(\cos t)) $
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00:42
Frank Lin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 4
The Chain Rule
Derivatives
Differentiation
Missouri State University
Harvey Mudd College
University of Michigan - Ann Arbor
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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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.
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