9–61. Evaluate and simplify y'.

y = tan ...

Question

9–61. Evaluate and simplify y'.

y = tan (sin θ)

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of Y equals the C1 X. A says it's sex squared X, X X B X X C negative X X, and D negative sec X X. Now, if you look at our function Y, you can tell that it's a composite function. We have an outer and inner function. How can we differentiate composite functions? Well, we can use the chain rule. Recall that if we have a function Y, OK, that's equal to U raised to the power of N, OK, where U is a function of X. Then the derivative of y with respect to X. Would be equal to the derivative of Y with respect to you multiplied by, let me fix this here. Multiplied by the derivative of u with respect to x. Now in our function here, Y equals the C of the tangent of X, OK, then. We know that U is equal to the tangent of X, so we can let you, sorry, be equal to the tangent of X. No, that tells us if we let you equal to an X, then Y is going to be equal to the 2 of you. Now let's differentiate both of these. If we differentiate you with respect to X, OK, the derivative of the tangent is the square of the c. So that's X squared X. That's the derivative of U or DU with respect to X. On the other hand, the derivative of Y with respect to you will be the derivative of se you with respect to you, which is gonna be secuan you. That's the derivative of the C. Now that we have DUDX and DUIDU applying the chain rule. That means the derivative of Y with respect to X or Y. Is going to be equal to secuan you. Sean you. Multiplied by 6 squared XDUD X. Now, to simplify, we can write uh what U is in terms of X and we know U equals 10 X. So really then, this is going to be 62 X multiplied by the C count of U, so that's the C count of 10 X. OK. Multiplied by the tangent of you. So that's the tangent of an X, OK? So in other words, then the derivative of Y is 62 X 6 X X. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

9–61. Evaluate and simplify y'.

y = tan (sin θ)

Key Concepts

Derivative

Chain Rule

Trigonometric Derivatives

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