Find the derivative of each function. y=log|1-x|

Name: Problem 33, Chapter 4: Calculus with Applications
Uploaded: 2020-01-30T15:56:08-08:00
Duration: 4 min 35 s
Channel: Robert Daugherty
Description: Find the derivative of each function. y=log|1-x|

Key Concepts

Logarithmic Differentiation

This concept involves differentiating logarithmic functions, particularly noting that the derivative of ln|u(x)|, where u(x) is a differentiable function and nonzero, is given by u'(x)/u(x). This approach works regardless of the absolute value because it absorbs any sign change in u(x) provided u(x) ? 0.

Chain Rule

The chain rule is fundamental when differentiating composite functions. It states that if you have a function y = f(g(x)), then the derivative y' is f'(g(x)) times g'(x). In the context of logarithmic functions, this rule helps in handling the inner function u(x) within the logarithm.

Handling Absolute Value in Differentiation

When differentiating a logarithm of an absolute value, such as ln|u(x)|, the absolute value ensures the argument of the logarithm is positive. The differentiation process remains consistent with the standard rule for logarithms since the presence of the absolute value does not affect the derivative as long as u(x) is nonzero.

Transcript

00:01 And section 4 .5, problem is number 33.

00:08 Again, everything in this section, derivatives dealing with logarithms.

00:12 What if we're presented with a function, y is equal to the log of the absolute value of 1 minus x? okay, keep in mind, this is log based in, not natural log.

00:24 So what are the things that are guiding us here? if y is equal to the log of x, we know that y prime is equal 1 over, over x times the natural log of the base, which is 10.

00:37 Okay, we do know that.

00:39 The other thing we know that is, if we have something that's a little more complicated, if y is equal to the log of f of x, then y prime is going to be f prime of x over f of x times the natural log of 10, which is the base.

01:03 So in this case, i've got the log of a function.

01:06 So what am i going to do using the chain rule here for the derivative? so y prime is going to be.

01:12 So here i've got f of x being this absolute value of 1 minus x.

01:16 So it's the absolute value of 1 minus x.

01:20 And then times the natural log of 10.

01:26 And then what is the derivative of the absolute value of 1 minus x? so how do you find the derivative of the absolute value of 1 minus x? okay, the trick for this is to really think of this definition of absolute value to think of this as 1 minus x squared to the one half.

01:52 So it's the square root of 1 minus x squared.

01:56 So how do you find the derivative of this expression? okay, the derivative of that expression is going to be one half.

02:10 Sorry, i'm running out of space here.

02:12 Let me come back up here.

02:13 So that is going to be one half and then so that's 1 minus x squared to the negative 1 half...

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