22–36. Derivatives Find the derivatives of th...

Question

22–36. Derivatives Find the derivatives of the following functions.

f(x) = csch⁻¹(2/x)

Accepted Answer

Hello. In this video, we are going to be finding the derivative of the given function F of X is equal to cossekin hyperbolic inverse of 7 divided by X. In order to approach this problem, let's quickly recall what the derivative of this function is. Now the derivative with respect to X of cosequent hyperbolic inverse of you. is equal to the negative 1 divided by the absolute value of u multiplied by the square root of 1 + u squared, and this will be multiplied by the derivative of u with respect to X. So what this definition is saying is that if we want to take the derivative with respect to X of the Kosic and hyperbolic inverse function. Whatever the U function is, which in this case is the angle to the trigonometric function, we are going to divide -1 by the absolute value of U, square that function, and multiply by that function's derivative. Now, let's just recall what is given to us in this problem. We are working with cosequent hyperbolic inverse of 7 divided by X. So based off of this definition, our function of U is going to equal to 7 divided by X. So, that means that for every word that we have a U, we are going to plug 7 divided by X into that function. Furthermore, we are going to need the derivative of you with respect to X. So by using the quotient rule, U is equal to -7 divided by X2. And so by using these definitions and the derivative of you, we will go ahead and get the derivative of the given function. So now if we take the derivative with respect to X, we will end up with -1. Divided by the absolute value of 7 divided by X multiplied by the square root of 1 + 7 divided by X quantity squared, then multiply by the derivative which is going to be -7 divided by X2. And now what we need to do is we need to simplify this derivative. So, first things first, a negative multiplied by negative is going to give us a positive term. In the numerator, we have 1 multiplied by 7, and that is going to leave us with 7. Now, in the denominator, we are going to have to multiply 7 divided by X with X2. That is going to leave us with the absolute value of 7 X. And this will be multiplied by the square root of 1 + 7 divided by X2, which is going to be 49 divided by X2. Now, to simplify the absolute value of 7 X, the absolute value of 7X is just going to be 7X. And notice that we have a multiple of 7 in both the numerator and denominator, so we can reduce these 7s to just be 1. So we are left with 1 divided by X multiplied by the square root of 1 + 49 divided by X2. Now, we can further simplify the denominator by rewriting X in the form of a square root. Recall that if we were to take the square root of X2, we get an individual X term. So we can rewrite X as the square root of X, and that will allow us to rewrite the derivative as 1 divided by the square root of X2, multiplied by the square root of 1 + 49 divided by X2. And by using our square root properties, we can multiply both of the square roots together, and that is going to leave us with one divided by the square root of X2 + 49. And this is going to be the simplest form of the derivative of the co-seeking hyperbolic function. And with that being said, the overall solution to this problem is going to be C. So I hope this video helps you in understanding how to approach this derivative, and we will go ahead and see you all in the next video.

22–36. Derivatives Find the derivatives of the following functions.

f(x) = csch⁻¹(2/x)

Key Concepts

Derivatives

Inverse Hyperbolic Functions

Chain Rule

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