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Find the derivative of the function. Simplify where possible.$ y = \cos^{-1} (\sin^{-1} t) $

$$y^{\prime}=\frac{-1}{\sqrt{1-\left(\sin ^{-1} t\right)^{2} \sqrt{1-t^{2}}}}$$

00:55

Frank L.

Doruk I.

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Faith A.

October 1, 2020

Missouri State University

Harvey Mudd College

Baylor University

University of Nottingham

Lectures

03:09

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So if this problem we're gonna be utilizing the derivative of inverse trig functions on this will be useful because we use inverse trig functions in math, they come up. So if we want to find the rate of change or the the slope of a tangent line of the graph, then we're gonna want to utilize in verse trig function identities on derivatives. So keep in mind that what we have in this case is the inverse coastline. So why equals the inverse coastline of the inverse sign of key? It could be acts, but in this case, it's teeth. All right, that's what we have. Remember that the derivative of the inverse coastline is one is negative. One over the square root of one minus X squared, where X is what's inside the parentheses. So now it will end up getting. Is that why crime is equal to negative one over the square root of one minus sine inverse of tea square? Yeah, And then, since we have to do change rule here, we're gonna multiplying that by the derivative of what's inside the parentheses, which that's another trick function in verse, trick function. That will be one over the square root of one minus. He squared because that's what happens when you take the derivative of the inverse sign of tea. Then we can multiply all this together to simplify it further. What will end up getting as a result is why prime being equal Thio negative one over the square root of one. Basically everything we see here. Uh, the only difference is we can just copy and paste this thing. Only difference is now we can multiply this bottom portion, uh, at the bottom. So we'll have times That's right here. And that will be our final answer for the derivative of why?

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