In Exercises 41–58, find dy/dt.

y = sin²(πt −...

Question

In Exercises 41–58, find dy/dt.

y = sin²(πt − 2)

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says to find the derivative of Y is equal to cosine squared of the quantity of 2 pi X minus 1 in quantity. So this problem we need to find the derivative of our function Y. So we're looking for Y, that's going to be equal to the derivative with respect to X of cosine squared. Of the quantity of 2 X. 1 in quantity. So in order to take this derivative, we're gonna need to recall several of our derivative rules. So, we're going to be taking the derivative of the cosine function. So recall that the derivative with respect to X. Of cosine of X. is equal to minus sign of X. We're also going to be um need to use our constant rule for derivatives, so we call for the constant rule for derivatives that that is the derivative with respect to X of a constant C is equal to 0. We're also going to need to use our power rule for derivatives to recall that we have the derivative with respect to X of X raised to the power n, that that is going to be equal to n multiplied by x raised to the quantity of n minus 1 in quantity. We're also going to need to use our concept multiple rule. We recall for that. We have the derivative with respect to X of the quantity of C multiplied by X raised to the power n in quantity, and that's going to be equal to C multiplied by N, multiplied by X raised to the quantity of n minus 1 in quantity. And finally, we're going to need to use our chain rule. So we call for the chain rule that we have the derivative with respect to X of F of G of X. And this is going to be equal to F of G of X. Multiplied by G of X. So we're gonna use all of these rules here to calculate our derivative. So that means that when we look at Y, it's going to be equal to. We'll first use our power rule and so we're going to be using our power rule on the cosine squared function. So we're going to multiply first our exponent, which is 2, and that gets multiplied by the quantity of cosine of the quantity of 2 X minus 1 in quantity, and that whole cosine function is raised to the power of 2 minus 1. We will also need to apply our chain rule to this, so we need to multiply this by the derivative with respect to X. Of cosine of the quantity of 2 pi X minus 1 in quantity. So this is going to be equal to 2 multiplied by cosine of the quantity of 2 X minus 1 in quantity, and now we have to multiply this by the derivative with respect to X of our cosine function. And so when you take a derivative of a cosine function, we'll get back a minus sign function. So we'll multiply this by minus sign. Of the quantity of 2 X minus 1 in quantity. But we also need to apply the chain rule to this as well, and so we'll need to multiply this by the derivative with respect to X of the quantity of 2 X minus 1 in quantity. So this is going to be equal to. Minus 2 multiplied by cosine of the quantity of 2 X minus 1 in quantity multiplied by the sine of the quantity of 2 X minus 1 in quantity, and now we need to multiply this by the derivative with respect to x of the quantity of 2 X minus 1. So when we take the derivative with respect to x of the quantity of 2 pi X, we're going to need to apply our constant multiple rule. So here we'll have our constant, which is 2 pi. Multiplied by the exponent, which is one. And then that gets multiplied by X, raised to the quantity of 1 minus 1 in quantity. Then we'll have minus the derivative of 1 with respect to X. And here we'll use our constant rule, and we're taking a derivative of a constant, which is equal to 0. So now we just need to simplify our expression here. And so that means that is going to be equal to minus or pi multiplied by cosine of the quantity of 2 x minus 1 in quantity multiplied by the sine of the quantity of 2 X minus 1 in quantity. So this here is the answer that we are looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

In Exercises 41–58, find dy/dt.

y = sin²(πt − 2)

Key Concepts

Chain Rule

Derivative of Sine Function

Power Rule

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