9–61. Evaluate and simplify y'.

y = e^si...

Question

9–61. Evaluate and simplify y'.

y = e^sin (cosx)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to calculate the derivative of the function Y is equal to E equates to the quantity of tangent of cosine X. We're given 4 possible choices as our answers. For choice A, we have minus E raised to the quantity of tangent of cosine X in quantity, multiplied by the quantity of sin X multiplied by squad of cosine X. For choice B, we have E raised to the quantity of tangent of cosine X in quantity, multiplied by the quantity of sin X multiplied by sequinsquared of cosine X. For choice C, we have minus E rates to the quantity of tangent of cosine X in quantity, multiplied by the quantity of cosine X multiplied by squared of cosine X. And for choice D, we have E rated to the quantity of tangent of cosine X in quantity, multiplied by the quantity of cosine X multiplied by squad of cosine X. Now, as you find the derivative of our function Y. In order to take this derivative, we're going to need to use our chain rule. So, to use our chain rule, we're going to identify you as being equal to tangent of cosine X. And that will allow us to rewrite Y as a function of use, that means that Y going to be equal toe raises to the power U. Now, if I take a derivative of Y, with respect to you, That it is going to be equal to E rays to the U as well, since the derivative of E rays to the U with respect to you is just E to the U. Now I'm also going to need to take a derivative of you. With respect to X, to apply our um chain rule. And so that derivative is going to be equal to, and here we're gonna have to actually apply the chain rule as well. The derivative of tangent is going to give me sequence squared. Of cosine X. And that's gonna be multiplied by the derivative of cosine X, which is going to be minus sine X. So we can now calculate the derivative of our function of Y with respect to X, so that's gonna be Y. This is going to be equal to the derivative of Y with respect to you. So that's gonna be E raised to the power U, but for you, I'm going to replace it with tangent of cosine X since we want our final answer as a function of X. So we'll have Y is equal to E raised to the. Power of tangent of cosine X. And then that is going to get multiplied by our derivative of you with respect to X, that's gonna be the quantity of squared. Of cosine X. Multiplied by Minus X. Now we're just going to rearrange the quantities here. So we'll pull the minus sign all the way out in front, so this is going to be equal to minus E raises to the quantity of tangent of cosine X. In quantity, that's gonna be multiplied by the quantity of, and here I'll just rearrange our two terms in our products, so we'll have sin x. Multiplied by Decon squared of cosine X. And so this is the answer that we are actually looking for. So that is the derivative of Y with respect to X. And if we compare our answer to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

9–61. Evaluate and simplify y'.

y = e^sin (cosx)

Key Concepts

Differentiation

Chain Rule

Exponential Functions

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