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# 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.

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## Lesson 3-6

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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

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