Find and simplify the derivative of the following functions.

Question

y = (3t−1)(2t−2)^-1

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to determine the derivative of the function F of X, which is equal to the quantity of 4 X plus 2 in quantity multiplied by the quantity of 3X minus 3 and quantity raised to the -1 power. We're given 4 possible choices as our answers. For choice A, we have F of X is equal to 3 divided by the quantity of X + 3 in quantity squared. For choice B, we have F of X is equal to 2 divided by the quantity of X + 1 in quantity squared. For choice C we have F of X is equal to -3 divided by the quantity of X minus 3 in quantity squared, and for choice D we have F of X is equal to -2 divided by the quantity of X minus 1 in quantity squared. Now, we're asked to determine the derivative of our function F of X, and to begin with, we're going to rewrite our function F of X. So we're going to rewrite that as F of X is equal to the quantity of 4 X plus 2. In quantity, divided by the quantity of 3 X minus 3. So writing it this way, we're going to be able to use our quotient rule in order to take the derivative. So recall for the quotient rule that we have the derivative with respect to X. Of the quantity of F of X. Divided by G of X. And this is going to be equal to. The quantity of G of X. Multiplied by F of X. Minus F of X. Multiplied by G of X. In quantity divided by The quantity of GFX. Squared So, using this definition, we're going to be calculating F of X. In our expression. So this is going to be equal to the quantity, and here we'll have our denominator, which is the quantity of 3 X minus 3. Multiplied by the derivative with respect to X of our numerator, which is the quantity of 4 X plus 2 in quantity, then we'll have minus our numerator, which is the quantity of 4 X. Plus 2, in quantity multiplied by the derivative with respect to X. Of our denominator, which is the quantity of 3 X minus 3 in quantity, and all of that is divided by Our denominator squared, which is the quantity of 3 X minus 3 in quantity squared. So now we just need to evaluate the derivatives in the numerator. So this is going to be equal to the quantity of 3X minus 3 in quantity multiplied by the derivative with respect to X of 4 X + 2, and that's going to give me 4, since the derivative of 4 X is 4 and the derivative of 2 is 0. Then I'll have minus the quantity of 4 X. Plus 2, in quantity, let's get multiplied by the derivative with respect to X of the quantity of 3 X minus 3, which is going to be equal to 3, hence the derivative of 3 X is 3, and the derivative of minus 3 is 0. And all of this is still being divided by the quantity of 3 X minus 3 in quantity squared. The performing the multiplication in the numerator, this is going to be equal to. 12 X -12. Minus 12 X. -6. And that whole quantity is divided by the quantity of 3 X minus 3 in quantity squared. So I'll simplifying the numerator, I see that um 12 X minus 12 X will sum to zero. And then I'll have 12, sorry, I have 12 minus 6. So this is going to give me -18, so this is going to be equal to -18, divided by, and here in our denominator, we're going to factor out a 3 squared. So we'll have 3 squared, multiplying the quantity of X minus 1 in quantity squared in the denominator. So, 3 squared is 9, and so minus 18 divided by 9, will give me minus 2. So this is going to be equal to -2. Divided by the quantity of X minus 1. In quantity squared. So this is the answer that we were looking for. And when we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Find and simplify the derivative of the following functions.
y = (3t−1)(2t−2)^-1

Key Concepts

Derivative

Product Rule

Quotient Rule

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