Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Given $P(x)=F(x) / S(x),$ where $F$ and $S$ are differentiable functions. Use logarithmic differentiation to obtain the usual quotient rule.

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Missouri State University

Harvey Mudd College

Baylor University

Idaho State University

Lectures

03:00

Given $P(x)=F(x) S(x),$ wh…

02:23

Suppose that $f$ and $g$ a…

04:54

Quotient Rule for the seco…

03:51

Consider the functions $f(…

01:31

Let $f(x), g(x),$ and $h(x…

eso We're gonna show the product started the quotient rule in this one and we're defining P as f over s. Okay, so what we want to start with with this problem is taking the natural audible sides. So we have natural log of p of X is equal to natural log of that quantity ffx over SFX and using the law of logs. What? We can establish this. That's the same. This natural log of F minus natural log of s That's from the law of locks. So then what we can do is we could take the derivative a beachside. Well, the derivative of natural log is one over the function itself, but with the chain role in their yet to multiply by the derivative of the function. So looking at this piece, then it's one over ethnics times have prime of X and then the last one is one over SFX times as prime of X But now is where the fun begins. What we need to do to solve for this function is ah to get p prime by itself. Well, we can only get p prime by itself. We multiply both sides by p of x times p of x. That way these things canceled and I can get P prime of X by itself Over here. Um, now, I switched to this color because p of X is equal to f of X over SFX. Yeah, So as I distribute that piece into each of these, what you'll notice is ah, in the first piece, ffx cancels. So what I'm left with is f prime of X, over S of X. All right. And then as I go to the next piece, what I have is s prime of X times f of X over. I have to SFX is, um but, uh, that's not the question rule. The question rule looks a little different. Well, all I need to do to get the quotient rule to be the same is noticed. I need a second SFX here. Well, that's only okay if I multiply the top by SFX as well. So you see how I'm getting the denominators to be the same. And the denominator, the way I have it written is SFX squared. Uh, there we go on. Then what I have is f prime of X s is not the derivative there minus. I just like to write it where f of X goes first and then as prime of X all over sf X squared And that is the quotient rule. So we did it. Kind of neat, Yeah.

View More Answers From This Book

Find Another Textbook

01:21

Evaluate the given integral.$$\int \frac{e^{\frac{-3}{x}}}{x^{2}} d x$$<…

03:46

For how long must $\$ 700$ be left on deposit at $6 \%$ compounded monthly t…

02:29

Use the properties of logarithms to find the derivative. Hint: It might be e…

01:20

Determine the derivative.$$f(x)=e^{-3 x}$$

03:02

Determine the equation of the tangent line at the indicated $x$ -coordinate.…

01:19

Determine $f^{\prime}(x)$.$$f(x)=\ln \left(x^{5}+12\right)$$

02:39

Evaluate the given integral.$$\int_{-1}^{0} x^{2}\left(2 x^{3}+1\right)^…

05:31

Find the partial derivatives with respect to (a) $x$ and (b) $y$.$$f(x, …

01:24

Compute the value of the given logarithm.$$\log _{5} 20$$

02:13

Consider the function defined by$$y=f(x)=\left\{\begin{array}{cc}x &…