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Differentiate the function.

$ F(t) = (\ln t)^2 \sin t $

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00:29

Frank Lin

00:52

Doruk Isik

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

01:13

Differentiate the function…

02:25

01:54

Differentiate.$$f(t)=\…

01:07

05:28

Find the derivative of the…

02:19

02:42

02:40

for this problem. We're wanting to graph the function. Um, in this case, the function is FFT is equal to the natural log of peace squared times The sign of team of the natural log of chief on the entire thing is squared so well, actually put the parentheses on the outside and that's time to sign of team. This is the problem that we're dealing with and we want to. Ultimately, our ultimate goal of this problem is to learn how to differentiate with log rhythmic functions. So what we'll do in this case is used the product rule. Um, so if the product rule, we take the first value and take its derivative. So when we do that, we'll end up getting the derivative of the natural log of teeth squared. So it'll be to natural log of tea times the inside because it'll be changeable. So times one over tea and then that's all going to be times the sign of team than what we have for the second portion is this is going to be plus, uh, the natural log of t squared. I'm the co sign of key since the derivative of scientist coastline key we see In both cases we can factor out the natural log of tea. So our final answer for this is going to be the natural log of tea times to sign T over tea, plus the natural log of tea coats. I'm t so ultimately, this is going to be our final answer. We factored out a natural log of tea here and here That ends up giving us two times the sine of t over tea and one natural log of T leftover times coastline T. This will be our derivative based on, um, using the product role and the chain role.

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