In Exercises 41–58, find dy/dt.

y = (1 + cos(...

Question

In Exercises 41–58, find dy/dt.

y = (1 + cos(2t))⁻⁴

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the derivative of Y equals parentheses 3 minus 2 multiplied by the cosine of 3 X all to the power of -3. Awesome. So it's important to note that 3 minus 2 multiplied by the cosine of 3 X is in parentheses, and the whole expression is to the power of -3. So ultimately, what we're trying to solve for is we're trying to find the derivative of this equation or function of Y. So now that we know that we're trying to perform the derivative for this particular problem, we must first recall. That D by DX of cosine of X is equal to negatives. Of X, and we also need to recall a few rules for differentiation. We need to recall our constant rule which states that D by DX of C, which C represents some constant value, is going to be equal to 0. We also need to recall the power rule which states that D by DX of X to the power of N is going to be equal to N multiplied by X to the power of N minus 1. And we also need to recall and use the constant multiple rule, which states that D by DX of C multiplied by X to the power of N is going to be equal to C multiplied by N multiplied by X to the power of N minus 1. And finally, we need to recall and use the chain rule which states that D by DX of F of G of X. Is going to be equal. 2 F. Of G of X. Multiplied by G of X. Fantastic. So with that in mind, we can go ahead and write that Y where this prime symbol indicates the first derivative of Y. So Y is equal to D by D X minus 2 multiplied by cosine of 3 X all to the power of -3. Fantastic. So now we can go ahead and write that Y is equal to -3 multiplied by 3 minus 2 multiplied by cosine of 3 X to the power of -3 minus 1 multiplied by D by DX of. So it's going to be multiplied by D by DX of once again entheses 3 minus 2 multiplied by cosine of. 3 X. Fantastic. So then we can go ahead and write that D by DX 3 minus 2 multiplied by cosine of 3 X is gonna be equal to 0 minus 2 multiplied by negatives of 3 X, multiplied by D by DX of 3 X. And this will be equal to 2 multiplied by sin of 3 X multiplied by 3 X to the power of 1 minus 1, which is going to be equal to 2 multiplied by of 3 X multiplied by 3, which is going to be equal to 6 multiplied by sign of 3 X, fantastic. Which will mean that we go ahead and write that Y is going to be equal to -3 multiplied by parentheses 3 minus 2 multiplied by cosine of 3 X, all to the power of -4 multiplied by 6 multiplied by sin of 3 X, which will mean when we simplify that our final answer will be Y is equal to negative. 18 multiplied by sin of 3 X divided by parentheses 3 minus 2 multiplied by cosine of 3 X all to the power of 4. Fantastic. And that is our final answer. Hooray, we did it. So to quickly recap and look at our original problem, That will mean, without a doubt, our final answer once again has to be that Y is equal to -18 multiplied by sin of 3 X, all divided by parentheses 3 minus 2 multiplied by cosine of 3 X all to the power of 4. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 41–58, find dy/dt.

y = (1 + cos(2t))⁻⁴

Key Concepts

Chain Rule

Trigonometric Derivatives

Power Rule

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