Finding Derivative Values

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Question

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = cot(πu/10), u = g(x) = 5√x, x = 1

Accepted Answer

Welcome back, everyone. Let F X equals the tangent of pi X divided by 8 and the G of X equals a root X. Find the value of the derivative of F of G of 4. A says it's 2 pi, B, C, 1/2 of pi, and D quarter of pi. Now notice here that we are trying to find the derivative of a composite function, OK? And if we recall, we can differentiate composite functions using the chain rule. Recall, the chain rule basically tells us that the derivative of a composite function F of G of X, OK. Is going to be equal to the derivative of FF G of X. Multiplied by the derivative of geoX. In other words, we differentiate the autoer function and multiply it by the derivative of the inner function. So in this case that means then that the derivative. Alright, well, we already have it, so I write it the way we have it. The derivative of FRG of 4. Is going to be equal to. The derivative of F of G of 4, OK, multiplied, that's the derivative of our other function multiplied by the derivative of G of 4. So here, if we're going to find this derivative, you can tell that there are probably a few things that we need, OK? First, notice that we'll need to figure out the derivative of GFX, and then we'll need to plug in. Uh, X equals 4, OK, to get the derivative of G of 4. Next, we'll need to find a derivative of F of X, OK, at X equals G 4, OK? And if we're gonna find it that X equals G 4, then we need to figure out G of 4, OK? So, you know, we have about 5 or 6 things that we need to find. So, let's go ahead and get started. Let's start with the derivative of G of X. Now here Fine here notice that. Given geoic, yeah. Which is equal to 8 multiplied by root X, OK. Then the derivative of G of X, OK, is gonna be equal to 8 multiplied by the derivative of root X, which is 1 divided by 2 root X. So this is gonna be 4 divided by root X. And if G of X is 4 divided by, sorry, the derivative of G of X is 4 divided by root X, then that means the derivative of G of 4 is gonna be equal to 4 divided by root 4. Is 4 divided by 2, which equals 2, OK? So if we make a note here, we have found the derivative of GF 4. Let's highlight that in our problem. Next, we need to find a derivative of F of X, OK? Now we know, OK, that FFX, as we're as as told in our problem, FF X. Is equal to the tangent of pi X divided by 8. So that means then that the derivative of FF X, to find the derivative, we'll need to figure out what we know about the tangent. I remember that the derivative of the tangent is the C square of X. So that means then, let me write that in the next line, OK? The derivative of FF X is going to be equal to the C2d of pi X divided by 8. Multiplied by the derivative of our inner function here, which is pi x divided by 8, the sans argument. And that is going to be equal to 1/8 of pi multiplied by c2 of pi X divided by 8. Now remember we already found out what G of the uh derivative of GF 4 is. But now we found F of X, OK? And we need to find it at X equals GF 4. What is GF 4? Well, we know that G of X equals 8 root X, so G of 4 is going to be equal to. 8 multiplied by root 4, that's 8 multiplied by 2, which equals 16. Thus, OK, this means that F of G of 4 is going to be equal to F of 16, which is going to be an 8 of pi multiplied by the c2 of i multiplied by 16 divided by 8. Now if we go ahead and simplify here. This is gonna be 1/8 of pi multiplied by the C squad of 2, OK? We know the c of 2 pi is 1, so this is gonna be 1/18 of pi multiplied by 1 squared, which is gonna leave us with 1/8 of pi, OK? Now, again, let's take a quick inventory of the information that we have here. What do we know? Well, we know that FG 4 is 1/8 of pi. We know that G 4 is 2. And remember, we're multiplying F of G of 4 by G 4. So now, since we have this information. This implies that FRGRG4, OK, then is going to be equal to. 8 of pi F G 4 multiplied by G of 4, which is 2. And this is equal to 14th of pie. Thus F of G 4 is equal to a 4th of pi. If we look back at our answer choices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = cot(πu/10), u = g(x) = 5√x, x = 1

Key Concepts

Chain Rule

Derivative of Cotangent Function

Square Root Function Derivative

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