1 / 8

Lesson 3-6

Lesson 3-6. Implicit Differentiation. Objectives. Use implicit differentiation to solve for dy/dx in given equations Use inverse trig rules to find the derivatives of inverse trig functions. Vocabulary.

saki
Download Presentation

Lesson 3-6

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lesson 3-6 Implicit Differentiation

  2. Objectives • Use implicit differentiation to solve for dy/dx in given equations • Use inverse trig rules to find the derivatives of inverse trig functions

  3. Vocabulary • Implicit Differentiation – differentiating both sides of an equation with respect to one variable and then solving for the other variable “prime” (derivative with respect to the first variable) • Orthogonal – curves are orthogonal if their tangent lines are perpendicular at each point of intersection • Orthogonal trajectories – are families of curves that are orthogonal to every curve in the other family (lots of applications in physics (example: lines of force and lines of constant potential in electricity)

  4. Derivatives of Inverse Trigonometric Functions d 1 d -1 ---- (sin-1 x) = ------------ ---- (cos-1 x) = ----------- dx √1 - x² dx √1 - x² d 1 d -1 ---- (tan-1 x) = ------------- ---- (cot-1 x) = ------------- dx 1 + x² dx 1 + x² d 1 d -1 ---- (sec-1 x) = ------------------ (csc-1 x) = ------------- dx x √ x² - 1 dx x √ x² - 1 Interesting Note: If f is any one-to-one differentiable function, it can be proved that its inverse function f-1 is also differentiable, except where its tangents are vertical.

  5. Example 1 Find the derivatives of the following: 1. arcsin (½ x) 2. arccos (x + 1) u = ½ x du/dx = ½ 1 du 1/2 ------------ ------------- = ------------------ (1 – x²) (1 – u²) (1 – (x/2)² u = (x + 1) du/dx = 1 -1 du 1 ------------ ------------- = ------------------ (1 – x²) (1 – u²) (1 – (x+1)²

  6. Example 2 Find the derivatives of the following: 3. arctan (x²) 4. arccot (x) u = x² du/dx = 2x 1 du 2x ------------ ------------- = ------------------ (1 + x²) (1 + u²) 1 + (x²)² u = x du/dx = ½ x-½ -1 du ½ x-½ 1/2 ------------ ------------- = ------------------ = ----------------- (1 + x²) (1 + u²) 1 + (x)² x (1 + x)

  7. Example 3 Find the derivatives of the following: 5. arcsec (ln x) 6. arccsc (xe2x) u = ln x du/dx = 1/x 1 du 1/x ------------ ---------------- = ------------------------ x(x² - 1) u (u² - 1) (ln x)(ln x)² - 1 u = xe2x du/dx = (2x + 1) e2x 1 du (2x + 1) e2x ------------ ---------------- = ------------------------ x(x² - 1) u (u² - 1) (xe2x)(xe2x)² - 1

  8. Summary & Homework • Summary: • Use implicit differentiation when equation can’t be solved for y = f(x) • Derivatives of inverse trig functions do not involve trig functions • Homework: • pg 233-235: 1, 6, 7, 11, 17, 25, 41, 47

More Related