9–61. Evaluate and simplify y'.

y = csc⁵...

Question

9–61. Evaluate and simplify y'.

y = csc⁵ 3x

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of the function Y equals 64th to X. A says it's 842 X multiplied by the tangent of 2 X. B. 46 cubed 2 X 2 X. C. 8 to the 4th 2 X square 2 X, and D 8 cubed 2 X 2 X. Now what's gonna help us to find the derivative of our function why? Well, if I rewrite why. As the second of 2 X. Raised to the 4th, OK, just write it in a more easily easily recognizable way. Then no, you can notice that we have one function in another. It's a composite function and to solve, we can apply the chain rule. What does the chain rule say? Basically, it tells us that if we have a function. If we have a function Y, that's equal to some composite function F of X raised to the power of N, OK. Then the derivative of Y is going to be equal to n multiplied by F of X to the power of n minus 1 multiplied by the derivative of F of X. In other words, we bring down the power, differentiate the inner function, and rewrite the outer function with one less in the power. Now how can we apply the chain rule here for why well. Since we know that our inner function F of X is equal to the C count of 2 X, OK. Then it's derivative, the derivative of a C count of 2 X is going to be equal to 2, 22X multiplied by the tangent of 2 X. Because the derivative of sec X is sec X1 X. and when we differentiate 2 X, we get 2. So that's what we would write outside the function. The C can function in itself is another composite function. So that's how we were able to get this term. Now that we have the derivative of the inner function, we can apply the chain rule. Which now tells us that the derivative of Y then is going to be equal to 4, we bring on the power multiplied by the C count of 2 X raise to the power of 34 minus 1. Multiplied by the derivative of the inner function, which is 22 X 12 X. Now, if we go ahead and simplify here, 4 multiplied by 2 is 8. we're gonna have the cube of the C count of 2 X. So we can write that as 8, a cube 2X. OK. Multiplied by sick. 2 eggs Multiplied by the tangent of tweaks. And now we can take it a step further to write set cubed 2 X multiplied by SE2X as Sec 4th 2 X. OK? In other words, SE 2 X-rays to the 4th part, multiplied by the tangent of 2 X. So the derivative of Y is 8642 X 2 X. If we look back at our answer choices, that means A is the correct answer. Thanks for watching everyone. I hope this video helped.

9–61. Evaluate and simplify y'.

y = csc⁵ 3x

Key Concepts

Differentiation

Chain Rule

Trigonometric Derivatives

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