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 1d -1 ---- (sin -1 x) = ---------------- (cos -1 x) = ----------- dx √1 - x²dx √1 - x² d 1d -1 ---- (tan -1 x) = ------------- ---- (cot -1 x) = ------------- dx 1 + x² d 1d -1 ---- (sec -1 x) = -------------- ---- (csc -1 x) = -------------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 1. arcsin (½ x) 2. arccos (x + 1) Find the derivatives of the following: 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 3. arctan (x²) 4. arccot (  x) Find the derivatives of the following: 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 5. arcsec (ln x) 6. arccsc (xe 2x ) Find the derivatives of the following: u = ln x du/dx = 1/x 1 du 1/x ------------ ---------------- = ------------------------ x  (x² - 1) u  (u² - 1) (ln x)  (ln x)² - 1 u = xe 2x du/dx = (2x + 1) e 2x 1 du (2x + 1) e 2x ------------ ---------------- = ------------------------ x  (x² - 1) u  (u² - 1) (xe 2x )  (xe 2x )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