In Exercises 41–58, find dy/dt.

y = (t tan(t)...

Question

In Exercises 41–58, find dy/dt.

y = (t tan(t))¹⁰

Accepted Answer

Hello there. Today we're going to 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 X multiplied by cotangent of 2 X all to the power of 11. Awesome. So it appears for this particular problem, we're ultimately trying to figure out what this derivative of this function of Y is. So now that we know that we're ultimately trying to figure out the derivative, we must first recall a few important details. We need to recall a few rules. So firstly, we need to recall that D by DX of cotangent of X is going to be equal to negative cocant. Squared of X, 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 of N minus 1. 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. We also need to recall and use the power rule, which the or not power rule, the product rule. So we also need to note that the product rule states that D by DX of F of X multiplied by G of X is going to be equal to G of X multiplied by F of X, and we also need to note that we need to add. F of X multiplied by G of X, fantastic. And lastly, we need to recall and use the chain rule, which, as we should recall, the chain rule states that D by DX of F of G of X. is going to be equal to F of G of X multiplied by G of X. Fantastic. So now we can go ahead and safely write that Y, which this prime symbol denotes the first derivative of Y, so Y is equal to D by DX of X multiplied by cotangent of 2 X to the power of 11, and we go ahead and write that Y is equal to 11 multiplied by X, multiplied by cotangent of 2 X. All to the power of 11 minus 1, multiplied by D by DX of X multiplied by cotangent of, drumroll, which is cotangent of 2 X, fantastic. So now we need to focus our attention on solving for F of X, and F of X is going to be equal to D by DX of X, which is going to be simply equal to 1. We now also need to find G of X and G of X is going to be equal to D by DX of cotangent of 2 X, which is going to be equal to. Negative cocant squad of 2 X multiplied by D by DX of 2 X, which is going to be all equal to negative cos squad of 2X multiplied by 2 X is the power of 1 minus 1, which will be equal to -2 multiplied by cosicant squad. Of 2 X. Fantastic. So once again, our answer for G of X is going to be -2 multiplied by Cosicant squad of 2 X. So now our next step is we need to apply the product rule to determine drumroll D by DX, so we need to use, once again, we're using the product rule to determine D by DX of X multiplied by cotangent of 2 X. Fantastic. So moving right along here, so therefore, we need to go ahead and write that D by DX of X multiplied by cotangent of 2 X is going to be equal to cotangent of 2 X multiplied by 1. Plus X multiplied by -2 multiplied by Cosicant squad of 2 X. Which is gonna be all equal to cotangent of 2 X minus 2. X multiplied by Cosicant square of 2 X. So therefore, without a doubt, Y will be equal to 11 multiplied by X, multiplied by cotangent of 2 X to the power of 10 multiplied by cotangent of 2 X minus 2. X multiplied by Cosicant squad of 2 X, and that is our final answer. Hooray, we did it. So looking at our problem itself, to quickly recap what we've learned, that will mean, without a doubt, our final answer has to be, once again, Y is equal to 11 multiplied by X, multiplied by cotangent of 2 X to the 10 multiplied by cotangent of 2X minus 2 X multiplied by CoCantsquad of 2X. 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 = (t tan(t))¹⁰

Key Concepts

Chain Rule

Product Rule

Trigonometric Derivatives

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