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

Question

f(x) = √(7 + x sec x)

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says determine the derivative of H of X, which is equal to the square root of the quantity of 5 minus X multiplied by cosequent of X in quantity. So this problem wants us to calculate the derivative of H of X. So we're gonna need to calculate each prime of X. Which is equal to the derivative with respect to X of the square root. Of the quantity of 5 minus x multiplied by cosequent of X in quantity. And so we're going to need to recall several derivative rules in order to take this derivative. First off, recall that the derivative with respect to X of the square root of X is equal to 1 divided by the quantity of 2 multiplied by the square root of X in quantity. And also recall that the derivative with respect to X of cosequant of X. is equal to minus cosequent of X multiplied by cotangent of X. We're also going to need to recall several of our derivative rules. The first rule that we're gonna need to recall is going to be our constant rule, and that is the derivative with respect to X of a constant C is equal to 0. We're also going to need to recall our product rule. And so that is going to be the derivative with respect to X. Of the quantity of F of X multiplied by GF X. And that is going to be equal to F of X. Multiplied by G of X. Plus F of X, multiplied by G of X. And then finally we also need to recall our chain rule, which is going to be the derivative with respect to X. Of F of G of X. And this is going to be equal to F of G of X. Multiplied by G of X. So, we're going to need to apply these rules to calculate this derivative. So, the first thing we're going to do is look at H X, and we're going to take the derivative of the square root function. And so that means H of X is going to be equal to 1 divided by. The quantity of 2 multiplied by the square root of the quantity of 5 minus x multiplied by cosequant of X in quantity. But we also have to apply our chain rules, so we'll need to multiply this by the derivative with respect to X. Of the quantity of 5 minus X multiplied by cosequant of X. In quantity So, taking this derivative, this is going to be equal to. The term out front remains the same, so we'll have 1 divided by the quantity of 2 multiplied by the square root of the quantity of 5 minus X multiplied by cosequent of X in quantity. And now we'll multiply this by the derivative specta X of the quantity of 5 minus. X multiplied by cosequent of X in quantity. So, taking the derivative of the first term there, we'll have the derivative of 5 with respect to X, and so I'm taking a derivative of a constant. So, by the constant rule that it's going to be equal to 0, then we'll have minus the derivative with respect to X. Of the quantity of X multiplied by cosequant of X. In quantity So, for that second term here, we're going to need to apply our product rule. So this is going to be equal to. Minus 1 divided by the quantity of 2 multiplied by the square root of the quantity of 5 minus X multiplied by the cosequent. Of X in quantity. Multiplied by the quantity, and here applying our product rule, we'll set F of X equal to X and G of X equal to cosequence of X. So we'll have F of X, so the derivative of X with respect to X is 1. That gets multiplied by G of X, which is cosequent of X. They'll have plus. F of X, which is X, multiplied by the derivative of G of X. And so the derivative with respect to x of cosequant of X we saw earlier was minus cosequant of X. Multiplied by cotangent of X. In quantity And so distributing the minus sign and simplifying, this is going to be equal to X multiplied by cosequent of X, multiplied by cotangent of X. Minus cosequant of X and that whole quantity is divided by the quantity of 2 multiplied by the square root of the quantity of 5 minus x multiplied by cosequant of X in quantity. And so this here is the answer that we are looking for for this problem. So I hope this explanation has been useful, and I'll see you in the next video.

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

f(x) = √(7 + x sec x)

Key Concepts

Derivative

Chain Rule

Product Rule

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