Find the derivative of the function. f(t)=e^a t sinb t |...

Name: Problem 19, Chapter 3: Calculus: Early Transcendentals, Metric Edition
Uploaded: 2022-09-14T16:50:09-08:00
Duration: 5 min 48 s
Channel: Elijah Dejonge
Description: Find the derivative of the function. f(t)=e^a t sinb t

Key Concepts

Exponential Function Differentiation

This concept deals with taking the derivative of exponential functions. Specifically, when differentiating an exponential function with a linear exponent, such as e^(at), the derivative results in a factor of 'a' multiplied by the original function, following the rule that d/dt[e^(at)] = a * e^(at). This rule is fundamental throughout calculus for handling growth and decay models.

Trigonometric Function Differentiation

This involves calculating the derivative of trigonometric functions like sine and cosine. For example, the derivative of sin(bt) is computed by applying the chain rule, resulting in b * cos(bt), where b arises from differentiating the inner function 'bt'. This principle is essential in solving problems involving periodic phenomena.

Product Rule

The product rule is critical when differentiating a function that is the product of two separate functions, such as an exponential component and a trigonometric component. The rule states that the derivative of a product f(t)g(t) is f'(t)g(t) + f(t)g'(t). This allows for the systematic differentiation of more complex expressions encountered in calculus.

Transcript

00:01 All right.

00:02 So for this problem, we want to find the derivative of this f of t here, e to the a t times sine of b t.

00:13 Okay.

00:14 And so we've got a clear, well, first of all, sorry, we've got f of t, right? that's our function.

00:22 So t is the variable that we're taking the derivative with respect to.

00:26 So this a and this b, we're just going to treat as constants.

00:29 We're just going to treat like their numbers.

00:31 We just don't know what they are okay so we've got e to the 80 times sign of bt okay so we've got two things being multiplied together so we're gonna use product rule alright so let's go ahead and get the product rule started and then we're gonna have to have another discussion okay so our f is gonna be the first thing so f is e the a t okay our g is going to be the second thing, sign of bt.

01:11 So if you look up here in product rule, on the left in the green here, we need f prime and we need g prime.

01:17 And this is where things start to get a little more complicated because e to the at is not one of our simple things that we know how to do.

01:27 We've actually got sort of an inside part, this a times t, and an outside part, this e.

01:36 So we're going to have to use chain rule.

01:39 So if we look up here at chain rule, what it's really telling us is we take the derivative of the outside, leave the inside the same, okay, and then we multiply by the derivative of the inside.

01:52 All right.

01:53 So what is the derivative of the outside? so the outside is e to the something, all right, to the u.

02:01 So what is the derivative of that? well, it's still just e to the u.

02:05 So we'll have e, okay, but instead of to the u, all right, if we look back up here, we're leaving g of x, we're leaving the inside the same.

02:17 So we're leaving at up here.

02:22 And now we need to multiply by the derivative of that thing that we left alone, okay, that at.

02:28 And what's the derivative of a .t? remember that we're doing this with respect to t.

02:34 So t is like x.

02:35 T is our variable.

02:38 A is just like a number.

02:40 So we would bring the power on the t down, or the power on the t is a 1.

02:45 Okay.

02:45 So we'd bring it down.

02:47 So we just get a times 1 times t to the 0.

02:51 So we just left with a.

02:53 So that's our derivative there.

02:57 We're going to have a similar thing going on here with g prime.

03:01 And just remember to, for both the product rule and channel rule, we're using f and g's.

03:09 We've actually got a lot of different fs and g's going on right now, right? this f of t was called the function.

03:15 We're forgetting about that for now.

03:16 All right...

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