In Exercises 13–24, find the derivative of y with respect to t...

Question

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

21. y = ln(cosh v) - 1/2 tanh²v

Accepted Answer

Welcome back, everyone. Find D Y divided by DU4 Y equals Lnci of u minus 1/2 squad of u. A cubed of you. B cost of you multiplied by 1 minus cos squared of you, C negative cubed of you, and D co cubed of you. For this problem we once find the derivative D or Y divided by DU. Notice that we have a difference of two functions. So, essentially, what we're going to do is differentiate the first function. A line of sin of you and we're going to subtract the derivative of the second one. We can also factor out the constant which is 1/2. So we're subtracting 1/2 multiplied by the derivative of c squared of u. Let's focus on the first derivative. We want to differentiate LM. Of sin of you. What we're going to do is simply use one of our rules, right? So the derivative of a land of sin of U is equal to us. Of you. And this is the first derivative that we are going to use. Now, for the other derivative, we want to find the derivative of cost squared of you. And here we want to apply the chain rule. Because we have a composite function, specifically, we have a square and cost of you. So according to the chain rule, we first of all differentiate our outer function with respect to the inner function. So we're differentiating cost squared with respect to cost, and this means that we're applying the power rule. We want to bring down the exponent, which is 2, and reduce the original exponent by 1 unit. So we're going to get 2. Cath of you. According to the chain rule, we're multiplying by the derivative of the inner function, which is the derivative of cause of you. And now the derivative of cost of you is negative squared of you, so we're going to get 2. Cause of you multiplied by negative. Coage squared off you. So now let's simplify. This is -2. Cos of you multiplied by cosse squared of you. Well, then, and now we have the required derivatives, we can substitute them into the original expression, so that the Y divided by the U is going to be equal to cost of you. -12 multiplied by -2 gives us 1. And then we're just adding cost of you. Multiplied by cosse squared of you. What we can see from here is simply factor out cost. Of you which leaves us with one plus. Cosa squared a few impres. And now 1 plus cosage squared of you. is basically equal to c squared of you. So we're going to get cost of you multiplied by. C squared of you. Which is equal to cost. Cubed of you and this corresponds to the answer to a. Thank you for watching.

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.21. y = ln(cosh v) - 1/2 tanh²v

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In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
21. y = ln(cosh v) - 1/2 tanh²v