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Find the derivative of the function.$ y = \cos \sqrt {\sin (\tan \pi x)} $

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00:49

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

David L.

March 29, 2022

heather, you're the best numerade tutor out there, you should work on more of the problems from chapter 3! you've helped a ton!

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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.

01:21

Find the derivative of the…

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02:47

we're going to find the derivative of this function, and I'm going to start by changing the notation just a little bit. I'm going to change the square root notation two exponents rotation. So we have Why equals co sign of the sign of tangent Pi X to the 1/2 power. Okay, so how many layers of chain do we have here? We have the inside most layer pi times x the next layer tangent the next layer sign the next layer 1/2 power and then the outermost layer co signs. We have five altogether. So we're going to be multiplying five derivatives together. So why Prime is the derivative of co sign the outermost one is negative sign. So have negative sign of everything that was inside it. I'm gonna go ahead and write it as a square root just for the sake of writing this so I don't have to change it later. Okay, Now we move on and multiply that by the derivative of the next layer. And the next layer was the 1/2 power. So we bring down the 1/2 and we raised the inside to the negative 1/2 the inside is sign of Tangent of Pi X So that's to the negative 1/2 power. Now we move to the next layer which is sign and the derivative of Sinus co sign. So we have co sign of Tangent of Pi X. Now we move to the next layer, which is tangent and the derivative of tangent is he can't squared So we have c can't squared of Pi X and then we moved to the innermost layer Pi x and it's derivative would be pie. Now let's see what we can do to simplify our answer. We have something with the negative exponents so we could bring that to the denominator. We can bring the two to the denominator from the 1/2 and we just have a single negative sign We can combine the negative sign and the pi toe lead us off So we have the opposite of pie times the sign of the square root of sign of tangent Pi X So we've taken care of the negative and the sign and the pie So far Now we're bringing the 22 the bottom from the 1/2 and then since we have a negative 1/2 power here. We'll bring that to the bottom and will make that a square root. Okay, so now we have the co sign of tangent Pi X on the top. And then we have the c can't squared of Pi X also on the top and nothing else combines. So that's it. That's our lovely answer.

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