In Exercises 41–58, find dy/dt.

y = ((3t − 4)...

Question

In Exercises 41–58, find dy/dt.

y = ((3t − 4) / (5t + 2))⁻⁵

Accepted Answer

Below 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 7 x minus 3 divided by 9 x + 4 all to the power of -6. Fantastic. So it appears for this particular prompt, we're asked to determine what the derivative of this equation of Y is. So now that we know that we're ultimately trying to take the derivative of this equation of Y. Our first step is to note that we can take our given function and we can write it as Y equals parentheses. 9 X + 4. Divided by 7 x minus 3 to 6. So first off, let us know we could take our given function, which once again is Y equals 7 X minus 3 divided by 9 X + 4 to the power of -6, and we could rewrite it as Y equals 9 x + 4 divided by 7 X minus 3 to the this whole expression to the power of 6. So now, our next step is we need to recall and use the following rules for differentiation. So let us recall the constant rule which states that D by DX of some constant value will be equal to 0. Let us also 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. And let us also recall and use the constant multiple rule which states that D by DX of some constant value C multiplied by X to the power of N will be equal to C, some constant value multiplied by N multiplied by X to the power of N minus 1. Let us also recall and use the quotient rule which states that D by DX of F of X divided by G of X is going to be equal to G of X multiplied by F of X minus F of X multiplied by G of X, all divided by G of X squad, fantastic. 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 to F of G of X multiplied by G of X. Fantastic. So now at this point, we need to solve for Y, which this prime symbol denotes the first derivative of our equation of Y, so we need to solve for Y. So Y is going to be equal to D by DX 9 X + 4 divided by 7 X minus 3 to the power of 6. So, when we start to perform this derivative, we will find that Y is equal to 6 multiplied by parentheses 9 X. Plus 4 divided by 7 X minus 3 to the power of 6 minus 1 multiplied by D by DX parentheses. 9 X, so when you take 9 X. So 9 X plus 4 divided by 7 X minus 3 once again in parentheses. So now at this point, we need to recall and note that for D by DX 9 X plus 4 divided by 7 X minus 3, let us state that F of X is going to be equal to 9 X + 4, and let us state that G of X is going to be equal to 7 X minus 3. So now we need to solve for F of X. So F of X is going to be equal to D by DX of 9 X +4, which is going to be equal. To 9 X to the power of 1 minus 1, which and then so it's gonna be 9 X to the power of 1 minus 1 + 0, which is going to be simply equal to 9 when we perform that derivative. Fantastic. And now we need to solve for G of X. So G of X is going to be equal to D by DX of 7 X minus 3, which is going to be equal to 7 X to the power of 1 minus 1 minus 0, which is going to be simply equal to 7 when we perform that derivative. So now at this point, we need to recall and apply the quotient rule to help us solve for D by DX of 9 X + 4 divided by 7 X minus 3. So when you take D by DX of 9 X + 4 divided by 7 X minus 3. Which is going to be equal to parentheses 7 X minus 3 multiplied by 9 minus 9 X + 4 multiplied by 7. All divided by 7 X or it's X minus 3, so 7 X minus 3 to the power of 2. So in an effort to save time, I'm going to skip a few steps, but essentially what we're going to do is we're going to take our current expression and we're going to write it in its most simple form. So we'll find out that D by DX, so we're gonna find that D by DX of 9 X + 4 divided by 7 X minus 3, it's going to be simply equal to negative. 55 divided by 7 X minus 3 to the power of 2. Fantastic. So, therefore, we will find that Y is going to be equal to 6 multiplied by parentheses 9 X + 4 divided by 7 X. So you need to take this divided by 7 X minus 3, and this expression will be to the power of 5 multiplied by negative. 55 divided by 7 X minus 3 to the power of 2. So when we simplify this expression, it will be equal to negative. 330 multiplied by 9 X + 4 to the power of 5. All divided by parentheses 7 X minus 3 to the power of 7. And that's it. That is our final answer for Hooray, we did it. We found our final answer for Y. We found our derivative. So, going back to look at the prom itself to quickly recap what we've learned, we know without a doubt now that our final answer has to be, once again, Y is equal to -330 multiplied by 9 X + 4 to the power of 5, all divided by 7 X minus 3 to the power of 7. 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 = ((3t − 4) / (5t + 2))⁻⁵

Key Concepts

Chain Rule

Quotient Rule

Negative Exponent Rule

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