Class Notes (839,590)
Mathematics (2,859)
MAT237Y1 (53)
Lecture

# 4.2 logs.pdf

3 Pages
121 Views

Department
Mathematics
Course Code
MAT237Y1
Professor
Dan Dolderman

This preview shows page 1. Sign up to view the full 3 pages of the document.
Description
Calc 1 Tues 10/7/08 4.2 - Derivatives of Log functions Well, if we’re going to ▯gure out what the derivative of ln(x) is, we might as well get cracking! d ln(x + h) ▯ ln(x) [ln(x)] = lim dx h!0 h 1 = lim [ln(x + h) ▯ ln(x)] h!0 h ▯ ▯ 1 x + h = lim ln h!0 h ▯ x ▯ 1 h = lim ln 1 + h!0 h x h Okay, now let’s do a substitution. Letxv =Notice that if we think of x as ▯xed, then when we take h ! 0, then v ! 0. This means 1 = lim ln(1 + v) v!0vx 1 1 = lim ln(1 + v) xv!0 v 1 1=v = lim ln(1 + v) xv!0 Oh, but we’ve seen that limit before! ...Or at least a variation of it. We’ve seen ▯ ▯ 1 x lim 1 + x!1 x which is exactly the same thing. What’s that limit? Why, it’s e! So, our limit is just 1 1 lim ln(e) = x v!0 x All right, so[ln(x)] =1, so long as x > 0. Also, for any base b, dx x d [log (x)] = 1 dx b xln(b) Example: ▯nd d [ln(2x + 3)]. dx Solution: This is just chain rule, so derivative of the outside woul, and then multiply 2x+3 that by the derivative of the inside, which is 2. So, the derivative will be 2 2x + 3 Since it’s so much easier taking the derivatives of sums and di▯erences rather than products and quotients (because you have to use those rules!), a lot of times, it’s much better to use log properties to split up the functions into sums and then take the derivative. 1 Example: ▯ ▯ d x sin(x) d ▯ ▯ ln 2 = ln(x ) + ln(sin(x)) ▯ ln(tan(x + 1)) dx tan(x + 1) dx 2 2 2 = 3x + cos(x)▯ 2xsec (x + 1) x3 sin(x) tan(x + 1) 2 2 3 2xsec (x + 1) = x + cot(x) ▯ tan(x + 1) Oh, and by the way, you can extend the derivative of log functions to d 1 lnjxj = ; x 6= 0 dx x This will become more important when we do integrals later on. But for now, let’s move on to
More Less

Only page 1 are available for preview. Some parts have been intentionally blurred.

Unlock Document

Unlock to view full version

Unlock Document
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.