In Exercises 41–58, find dy/dt.

y = (1 + tan⁴...

Question

In Exercises 41–58, find dy/dt.

y = (1 + tan⁴(t/12))³

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 parentheses 7 + tangent to the power of 5 of x divided by 10. Parentheses all to the power of 4. So it appears that in parentheses, so 7 + tangent to the power of 5 of X divided by 10, which is in parentheses all to the power of 4. Awesome. So it appears for this particular problem, we're ultimately trying to take this equation of Y and we're trying to find the derivative of it. So now that we know that we're ultimately trying to figure out the derivative. Our first step is we need to recall that D by DX of Tangent of X is going to be equal to c 2d. Of X We also need to recall and use the following rules for differentiation. We need to recall the constant rule which states that D by DX of C, which C is some constant value, is equal to 0. We also need to recall and use the power rule which states that D by DX of X to the power of N is equal to N multiplied by X to the power of N minus 1. We also need to recall and use the constant multiple rule which states that D by DX of some constant value multiplied by X to the power of N will be equal to some constant value multiplied by N, multiplied by X to the power of N minus 1. And finally, we also need to recall and use the chain rule which states that D by DX of F of G of X is gonna be equal to F of G of X, multiplied by G of X. Fantastic. So now we need to solve for a Y, where Y where this prime symbol indicates the first derivative. So we need to recall that once again we're trying to solve for Y ultimately. So Y is going to be equal to D by DX. 7 plus tangent to the power of 5 of x divided by 10. To the and then this expression will be to the power of 4, fantastic. So now, our next step is we need to take Y is equal to 4 because we're trying to take the derivative and we're also simplifying as we go. So we'll find that when we do our first simplification and start to perform our derivatives, we will get that Y is equal to 4 multiplied by 7 plus tangent to the power of 5 of X divided by 10. And this expression will be to the power of 4 minus 1, multiplied by D by DX of 7 plus tangent to the power of 5 of X divided by 10. Fantastic So now, moving right along here. Our next step now is we need to take D by DX of 7 plus tangent to the power of 5 of x divided by 10 will be equal to 0 + 5 multiplied by tangent to the power of 5 minus 1 of X divided by. 10 multiplied by D by DX of, tangent of X divided by 10, fantastic. And now, our next step is we need to take D by DX of 7 plus. Tangent to the power of 5 of X divided by 10. is going to be equal to 5 multiplied by tangent to the power of 4 of X divided by 10. And then we're gonna take this and we're gonna multiply it by c can squad of X divided by 10, multiplied by D by D X X divided by 10. So once again, we're trying to simplify and take derivatives as we go. These we're trying to find our final derivative for Y. So now, moving right along, we can now go ahead and write that D by DX of 7 plus tangent of 5, so tangent to the power of 5. So we're taking D by DX of 7 plus tangent to the power of 5 of X divided by 10. It's gonna be equal to 5 multiplied by tangent to the power of 4 of X divided by 10, multiplied by C2 x divided by 10, multiplied by 1 divided by 10 or 1/10. And finally, continuing onward here. We also can go ahead and write that D by DX 7 plus tangent to the power of 5 of X divided by 10 is going to be equal to 1/2 multiplied by tangent to the power of 4 of X divided by 10, multiplied by c2 of X divided by 10. Fantastic. So moving right along here. So now you can go ahead and write that Y is going to be equal to 4 multiplied by parentheses 7 plus tangent to the power of 5 of X divided by 10. And this expression will be to the power of 3, multiplied by parenthesis of 1/2 multiplied by tangent to the power of 4 of X divided by 10. Multiplied by C can squad of X divided by 10. So finally, when we simplify one last time, we will find that Y is going to be simply equal to 2 multiplied by tangent to the power of 4 of X divided by 10 multiplied by C2 x divided by 10, multiplied by parentheses 7 plus tangent to the power of 5 of X divided by 10. And this expression will be to the power of 3. So, when I say this expression, I mean in parentheses 7 plus tangent to the power of 5 of X divided by 10 will be all to the power of 3. Fantastic, and that is our final answer. Hooray, we did it. So going back to look at our problem itself to quickly recap what we've learned together, once again, our final answer, without a doubt, will have to be the following. So to quickly restate what we've learned together. So once again, our final answer, without a doubt has to be drum roll. Y is equal to 2 multiplied by tangent to the 4 of X divided by 10. Multiplied by c squad of x divided by 10, multiplied by parentheses 7 plus tangent to the power of 5 of X divided by 10 to the power of 3. So once again, it's important to note that in To see 7 + tangent to the power of 5 of X divided by 10, it's going to be to the power of 3. And that's it. That is our final answer. 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 + tan⁴(t/12))³

Key Concepts

Chain Rule

Derivative of Trigonometric Functions

Power Rule

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