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Differentiate the function.

$ y = \ln (e^{-x} + xe^{-x}) $

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00:56

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

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.

00:55

Differentiate the function…

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Differentiate the followin…

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01:02

In this problem, we are to find a derivative of natural log of 2 negative x plus x times negative x since, inter negative x are common. Here we can find this 1 as natural log of each negative x, 1 plus x. Now we have 2 functions in multiplication form and using properties of logatfunctions. We can write this 1 as natural log of etithenegative x, plus natural log of 4 plus x. We have natural log of each negative x. We know that this is nothing but negative x. So we basically are trying to find derivative of negative x, plus natural log of 1 plus x plus y pine is then negative, 1 plus 1 over 1 plus x. Now, if we multiply and divide 1 plus x, if you divide and multiply first term by 1 plus x, we find it a negative 1 minus x, plus 1 divided by 1 plus x. These were cancel out and the answer is the negative x, divided by 1 plus x,

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