9–61. Evaluate and simplify y'.

y = (sin...

Question

9–61. Evaluate and simplify y'.

y = (sin x / cos x+1)^1/3

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to find the derivative of the function Y, which is equal to the quantity of 2 multiplied by tangent of X, divided by the quantity of 1 minus sin of x in quantity, and that whole fraction is raised to the 12 power. So this problem wants us to find the derivative of our function Y, and in order to do that, we're going to need to apply our chain rule. So we call for the chain rule that we need to identify our inner and outer functions. So our inner function u is going to be equal to 2 multiplied by tangent of X divided by the quantity of 1 minus sine of x. And this will allow us to rewrite our our function Y as a function of you, that means that Y going to be equal to you raised to the one half power. So this is going to be our outer function. Now, recall in order to apply our chain rule, we need to find the derivative of U with respect to X. And this is going to be equal to the derivative with respect to X of the quantity of 2 multiplied by tangent of X, divided by the quantity of 1 minus sine of X. So, in order to take this derivative, we're gonna need to use our quotient rule. So this is going to be equal to, and here we call for your quotient rule that we'll have our denominator, which is the quantity of 1 minus sine of X in quantity, and that gets multiplied by the derivative of our numerator. So when I take a derivative with respect to X of 2 multiplied by tangent of X, I get 2 multiplied by squad of X, then we'll have minus our numerator, which is 2 multiplied by the tangent of X. Then that gets multiplied by the derivative of our denominator. So, when I take a derivative with respect to X of 1 minus sine of X, I get minus cosine of X. Then this whole quantity is divided by our denominator squared, which is the quantity of 1 minus sine of X in quantity squared. So we can simplify this expression. So this is going to be equal to. I can factor out of 2 out of both terms in our numerator. So we'll have 2 multiplying the quantity, and here we'll just multiply everything else out in the numerator. So we will have sequent squared of X minus. Sign of X multiplied by sequent squad of X, then we'll have plus, and when I multiply tangent of X with cosine of X, I get sin of X. In quantity divided by The quantity of 1 minus sine of X in quantity squared. We also need to calculate the derivative of Y with respect to you, which we'll need to use in our chain rule later. So this is going to be equal to the derivative with respect to you, of the quantity of you raised to the 12 power. So here applying our power rule, this is going to be equal to 1/2 multiplied by U raised to the minus 1/2 power, which we can rewrite as being equal to 1 divided by 2 multiplied by U raised to the 1/2 power. So, now we have all the information that we need to calculate our derivative of our function Y with respect to X. We'll label that as Y. And so recall for the chain rule that Y is going to be equal to the derivative of Y with respect to U, multiplied by the derivative of U with respect to X. So, here we're just going to substitute in the quantities that we have calculated. So, this is going to be equal to. For the derivative of y with respect to you, we will have 1 divided by the quantity of 2 multiplied by the quantity, and here we'll have for you, we're going need to substitute in back our original expression for you in terms of X because we want our final answer as a function of X. So that is going to be 2 multiplied by tangent of X divided by the quantity of 1 minus sine of X in quantity, and this whole quantity is raised to the 12 power. We then need to multiply this by the derivative of U with respect to X, which is 2 multiplied by the quantity of sequent squad of X minus sine of X multiplied by squad of X. Plus sign of X. In quantity divided by The quantity of 1 minus sine of x in quantity squared. So now we just need to simplify this expression. I noticed that the factor of 2 in the numerator will cancel out with the factor of 2 in the denominator. So this is going to be equal 2. On the numerator, I'm going to have sequent squared of X. Minus sign of X multiplied by squad of X plus sign of X. Net quantity is divided by. The quantity of, and in the denominator, I'll have two rates to the 1/2 power, which is just the square root of 2. That's gonna get multiplied by the quantity of tangent of X. Multiplied by the quantity of 1 minus sine of X in quantity cubed. In quantity raises to the 1/2. So here we have moved the quantity of 1 minus sine of X in quantity squared inside the parentheses that are being raised to the 12 power. So that gives me the quantity of 1 minus sign of X in quantity raised to the 4th power, but that is being divided by a factor of 1 minus sin of x, so that leads me to just the quantity of 1 minus sign of X in quantity cubed. So, this is the answer that we're actually looking for for this problem. So this is the answer to the problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

9–61. Evaluate and simplify y'.

y = (sin x / cos x+1)^1/3

Key Concepts

Derivative

Chain Rule

Quotient Rule

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