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) = u + 1/cos²u, u = g(x) = πx, x = 1/4

Accepted Answer

Welcome back, everyone. In this problem, let F of X equals 2 X plus 1 divided by the square of sin X and G of X equals pi X divided by 2. Find the value of F of G of 1/2. A says it's negative pi divided by 2, B negative pi, C pi divided by 2, and D says it's pi. Now what are we really trying to figure out here? Well, this means, or this is the derivative of the composite function F of G X evaluated at X equals 1.5. So if we're differentiating our composite function, we'll need the chain rule. Recall that the chain rule tells us that the derivative of a composite function. F of GFX. OK. Is going to be equal. OK, equal to the derivative of F of G of X, OK, multiplied by the derivative of G of X. In other words, we differentiate the other function and multiply it by the derivative of the inner function. So applying the chain rule, that means, OK, the derivative of F of G of 12 value we're looking for. Is going to be equal to F of G2, OK, multiplied by G2. So in order for us to differentiate, there are a few things that we'll need based on this statement. First, notice that we'll need G of half. Now we know, OK, that G of X equals pi x divided by 2. So G of 1/2 is going to be equal to 1/2 of pi multiplied by 1/2, which equals 4th of pi. OK, so now we have this expression. Next, we need to find G of a half, but to do that, we need to find the derivative of G of X G X. No, If we do that, if we differentiate G of X, OK, then we're going to be finding the derivative of pi X divided by 2, which is going to be pi divided by 2, OK? So if the derivative of G of X is pi divided by 2, then G of 1/2 is going to also be equal to pi divided by 2 because notice here that there is no X term in the derivative. The derivative is a constant, so it's going to stay the same. The value of G of X at any X will be a half of pi. Know that we have that. OK, let's mark it off in our term here. Next, we need to find F of G2. To find that we need to differentiate FFX. Now, given F of X, which is equal to 2 X plus a coi can square of X, then the derivative of FF X is going to be the derivative of that expression. Now how do we differentiate that? Well, if we think about it here, first we'll differentiate to X, which will give us 2, and then we're going to differentiate coc2 X. Again, this is another composite function. So by the chain rule, it's going to be 2 multiplied by the core C is 2 minus 1 of X, multiplied by the derivative of its inner function, which would be the corecant of X. Again, if we differentiate that, OK. Now, we're gonna have 2 + 2 cosicant X multiplied by negative cosicant X. OK. Multiplied by the cotangent of X, and now when we simplify here we get the derivative of F of X to be 2 minus 2 cosic square of x multiplied by the cotangent of X. And if this is the derivative of F of X, OK, then this would imply. That F of G2, OK. Which is gonna be F of 1/4 of pi, OK. It's not gonna be equal to 2 minus 2 multiplied by the coi can squared off pi divided by 4. Multiplied by the cotangent of pi divided by 4. No, if we go ahead and simplify here, so this is gonna be 2 minus 2 multiplied by root 2 squared, OK? And that's because the coequant is the reciprocal of a sign. So the sign, OK, would be 1 divided by root 2, the sign of a 4th of pi. That's why the cosant is root 2. So that's 2 multiplied by root 2 squared multiplied by the cotangent of 1/4 of pi, which equals 1, OK? No, We'll go ahead and simplify again. So that's gonna be 2 minus 2 multiplied by 22 multiplied by 2 is 4, and 2 minus 4 equals -2. Thus F of G1/2 equals -2. So now what do we know? We know F of G. We know G2, and that means then that we can solve for our composite function because no, that means F of G sorry F of G12. Is going to be equal to F G 1/2 multiplied by G 2 I divided by 2, and this will be equal to negative pi. Therefore, our answer is negative pi and when we look back at our answer choices, that means B 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) = u + 1/cos²u, u = g(x) = πx, x = 1/4

Key Concepts

Chain Rule

Derivative of Trigonometric Functions

Substitution

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