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Use logarithmic differentiation to find the derivative of the function.

$ y = (\sin x)^{\ln x} $

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01:12

Frank Lin

Doruk Isik

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

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.

02:10

Use logarithmic differenti…

01:24

00:56

01:10

02:38

01:18

01:39

in this problem, we are learning how to find the derivative of a function using log rhythmic differentiation. And this is very helpful because we can simplify exponents very easily using log log properties. Excuse me. So we have the function. Why equal society of X rays to the natural log of X and this looks really complicated, but it's not going to be extremely complicated if we use log rhythmic differentiation. So the first thing we're going to do is take the natural log of each side. You would get the natural log y equals the natural log of the sign of X rays to the natural log of X. But using the properties of logs, we can pull out that exponents so we would get the natural log of y equals the natural log of X are exponents times the natural log of the sign of X, basically what's left over. So then we can derive each side of the equation. And for our right side, we have to use product rule as well as chain role. So we're going to get one over. Why times do I d X equals one over x times the knock tra log of sine X plus the natural log of X times. Now this is our changeable component one over the sine of x times, the co sign of X. And then we can simplify a little bit and we can move this. Why to the other side. So will multiply by Why and I did a little bit of simplification with our trig and metric function here will get d y d X equals y times one over x times a natural log of the sign of X plus the natural log of x times the co tangent of X. But we know what? Why is that? That was what what we started with. So we can just substitute that function in and we'll find that d Y d X equals the sign of x rays to the natural log of X times this entire quantity the natural log of the sign of X over X plus the natural log of x times, the co tangent of X and that is our derivative. So I hope this problem helped you understand a little bit more about the motivation behind log rhythmic differentiation and how we go through the process of applying it to a function and finding its derivative

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