Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this document? Why not share!

- Lesson 22: Optimization Problems (w... by Matthew Leingang 4595 views
- Gluing together Blackboard, Faceboo... by Matthew Leingang 26150 views
- Forgotten calculus pdf by Educationtempe78 105 views
- Electronic Grading of Paper Assessm... by Matthew Leingang 2360 views
- Streamlining assessment, feedback, ... by Matthew Leingang 1174 views
- Lesson 16: Derivatives of Exponenti... by Matthew Leingang 31737 views

4,180 views

Published on

No Downloads

Total views

4,180

On SlideShare

0

From Embeds

0

Number of Embeds

1

Shares

0

Downloads

71

Comments

0

Likes

1

No embeds

No notes for slide

- 1. Solutions to Worksheet for Sections 3.1–3.3 Derivatives of Exponential and Logarithmic Functions V63.0121, Calculus I Summer 2010 Find the derivatives of the following functions. 2 −3x 1. y = e2x Solution. We have dy 2 d 2 = e2x −3x 2x2 − 3x = e2x −3x (4x − 3) dx dx 2. y = 62x Solution. We have y = (ln 6) · 62x · 2 = (2 ln 6)62x 3. y = ln(x3 + 9) Solution. We have dy 1 d 3x2 = 3 x3 + 9 = 3 dx x + 9 dx x +9 4. y = log3 ex Solution. By brute force we have 1 1 y = · ex = (ln 3) · ex ln 3 But slightly more elegantly, we notice that ln ex x log3 ex = = ln 3 ln 3 dy 1 So = makes a lot of sense. dx ln 3 1
- 2. 5. y = log10 3θ2 −θ Solution. dy 1 2 −1/2 1 2 = log10 3θ −θ · θ 2 −θ · (ln 3) · 3θ −θ (2θ − 1) dx 2 (ln 10)3 ln 3 2θ − 1 = · 2 ln 10 log10 3θ2 −θ There’s some simpliﬁcations we could do before diﬀerentiation, however. 2 −θ ln 3θ θ2 − θ ln 3 ln 3 2 θ 2 −θ log10 3 = = = θ −θ ln 10 ln 10 ln 10 So ln 3 y= · θ2 − θ ln 10 ln 3 2θ − 1 y = · √ ln 10 2 θ2 − θ 6. y = sin2 x + 2sin x Solution. y = 2 sin x cos x + 2sin x · ln 2 · cos x Use logarithmic diﬀerentiation to ﬁnd the derivatives of the following functions. 7. y = x x2 − 1 Solution. We have 1 ln(x2 − 1) ln y = ln x + 2 1 dy 1 1 2x = + · 2 y dx x 2 x −1 dy 1 x = x x2 − 1 + 2 dx x x −1 8. y = (x − 1)(x − 2)(x − 3) 2
- 3. Solution. We have 1 ln y = (ln(x − 1) + ln(x − 2) + ln(x − 3)) 2 1 dy 1 1 1 1 = + + y dx 2 x−1 x−2 x−3 dy 1 1 1 1 = (x − 1)(x − 2)(x − 3) + + dx 2 x−1 x−2 x−3 x(x − 1)3/2 9. y = √ x+1 Solution. We have 3 1 ln y = ln x +ln(x − 1) − ln(x + 1) 2 2 1 dy 1 3 1 1 1 = + · − · y dx x 2 x−1 2 x+1 dy x(x − 1)3/2 1 3 1 1 1 = √ + · − · dx x+1 x 2 x−1 2 x+1 3

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment