Find the derivatives of the functions in Exercises 19–40.

Question

g(t) = (1 + sin(3t) / (3 − 2t))⁻¹

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says determine the derivative of H of T, which is equal to the quantity of 2 plus cosine of 4T in quantity, divided by the quantity of 4 minus 3T in quantity, and that whole fraction is raised to the -1 power. So we need to find the derivative of H of T. So before we begin to take the derivative, we need to just rewrite our function H of T. So H of T. Can be rewritten as being equal to the quantity of 4 minus 3T. In quantity divided by the quantity of 2 plus cosine of 4 T in quantity. So here we just applied the -1 power to our fraction. So now we can take the derivative, and so we're gonna be looking for H of T, which is going to be equal to the derivative with respect to T. Of the quantity of 4 minus 3 T. In quantity divided by the quantity of 2 plus cosine of 4 T in quantity. And so in order to take this derivative, we're going to need to recall several of our derivative rules. We notice that we're taking a derivative of the fraction, so we're going to want to use our quotient rule. So recall for the quotient rule that we will have the derivative with respect to T of the quantity of F of T divided by G of T. In quantity, and this is going to be equal to the quantity of GFT. Multiplied by F of T. Minus F of T. Multiplied by G of T in quantity, divided by. The quantity of GFT squared. We're also going to need to apply our chain rule. So, recall for the chain rule that we have the derivative with respect to T of F of G of T. And this is going to be equal to F of G of T. Multiplied by G of T. We're also going to need to use our constant rule for derivatives, so if we call for the constant rule that will have the derivative with respect to T of a constant C, and that's going to be equal to 0, and we're also going to need to use our constant multiple rule. And so recall for that, we have the derivative with respect to T of the quantity of C multiplied by T raised to the power n. And that is going to be equal to C multiplied by N, multiplied by T raised to the quantity of N minus 1 in quantity. And then finally, we'll need to recall the derivative with respect to T of cosine of T. And that it's going to be equal to minus sign of T. So, we're going to use all these rules in order to calculate our derivative here. And since we're going to be using the quotient rule, it's going to be good to identify the functions FFT and GFT in that rule. So here we're gonna identify FFT. To be equal to 4 minus 3T, and we're gonna identify GFT to be equal to 2 plus cosine of 4 T. So, we'll need to calculate F of T. And that's going to be equal to the derivative with respect to T. Of the quantity of 4 minus 3 t in quantity. And so taking this derivative, this is going to be equal to. For the first term, we'll have the derivative of 4 with respect to T, and by our constant rule, that it's going to be equal to 0. Then we'll have minus the derivative with respect to TF 3T. And so here we'll apply our constant multiple rule. So we'll have the constant 3 multiplied by the exponent, which is 1, multiplied by T raised to the quantity of 1 minus 1 in quantity. And when we simplify this expression, this is going to be equal to minus 3. For G of T, that is going to be equal to the derivative with respect to T of the quantity of 2 plus cosine of 4 T. In quantity. And so taking this derivative, this is going to be equal to, for the first term, we'll have the derivative of 2 with respect to T. Again, taking a derivative of constant, that's going to be equal to 0. Then we'll have plus. We'll be taking the derivative with respect to T of our cosine function, which will give us minus sign of 4 T, but we also need to apply our chain rule. And so, we'll need to multiply this by the derivative with respect to T of 4T. So this is going to be equal to minus sign of 40. Multiplied by, and here when we take the derivative with respect to T of 40, we'll need to apply our constant multiple rule again. And so we'll have a constant 4 multiplied by the exponent, which is 1, multiplied by T, raised to the quantity of 1 minus 1 in quantity. And so simplifying this expression, this is going to be equal to -4, multiplied by sine of 4 T. And so now we can use our quotient rule in order to calculate the derivative. And so, we're going to calculate each prime of T now, and this is going to be equal to. From our quotient root, this is going to be the quantity of GFT, which is the quantity of 2 plus cosine of 4 T. In quantity, multiplied by F T, which is minus 3. Then we'll have minus F of T, which is the quantity of 4 minus 3T in quantity, multiplied by G of T, which is -4 multiplied by sine of 4 T in quantity. And all that is divided by. GF T squared, so it's gonna be the quantity of 2 plus cosine of 4 T. In quantity squared. So now we just need to simplify this expression. So, multiplying everything out in the numerator, this is going to be equal to the quantity of -6, minus 3, multiplied by cosine of 40. Then we'll have + 16, sign of 40. Then we'll have minus 12 T multiplied by sine of 4 T. In quantity, and that is still being divided by the quantity of 2 plus cosine of 4 T. In quantity squared. So we can factor out a negative sign of the numerator. And so this is going to be equal to. Minus the quantity of 6 + 3 multiplied by cosine of 40. -16 multiplied by sin of 40. Plus 12t multiplied by sine of 4 T. In quantity divided by The quantity of 2 plus cosine of 4 t in quantity squared. So this here is the answer that we were looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Find the derivatives of the functions in Exercises 19–40.

g(t) = (1 + sin(3t) / (3 − 2t))⁻¹

Key Concepts

Chain Rule

Quotient Rule

Trigonometric Derivatives

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