Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

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1 Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here are some formal efinitions from tat section: A Secant Line is a line connecting two points on a curve of a function f(x). Te slope of a secant line, also known as te average rate of cange on a given interval [a, x] is f(x) f(a) m sec = x a Also, we remember tat te Tangent Line is te line tat intersects te curve of a function f(x) at exactly one point, a. Also, te slope of te tangent line, or instantaneous rate of cange is te limit of te slope of te secant lines: m tan x a f(x) f(a) x a Te tangent line at x = a is te unique line troug te point (a, f(a)) wit slope m tan. Its equation is y f(a) = m tan (x a) Example 1: Let f(x) = 16x x. a) Fin te slope of te tangent line at te point (1, 80) b) Fin te equation of te tangent line. Solution a). m tan x 1 f(x) f(1) x 1 ( 16x x) 80 x 1 x 1 16(x 5)(x 1) x 1 x 1 = 64 b) We foun te slope in a), so we nee to plug te point into point-slope form: y 80 = 64(x 1) 1

2 2 Te Derivative Te efinition of te slope of te tangent line can be written ifferently. Assume we want to fin te secant line on te interval x = [a, a + ], were is a small value. Ten, te slope of te secant line at point x = a can be written as m tan f(a + ) f(a) We efine te Dervative of a function f(x) as te slope of te tangent line. We write te erivative of f(x) as f (x), spoken as f prime of x. Generally, te erivative of a function f at any point x is given as f (x) f(x + ) f(x) Specifically, te erivative at a point x = a is given by f (a) f(a + ) f(a) If f (a) exists, ten f is sai to be Differentiable at te point x = a Example 2: Fin te erivative of f(x) = 16x x. Solution: f (x) f(x + ) f(x) f(x) f(x+) { }} { { }} { 16(x + ) (x + ) ( 16x x) 16(x 2 + 2x + 2 ) + 96x x 2 96x ( 32x ) ( 32x ) = 32x + 96 Tis f (x) is a function of x tat tells us te slope of te tangent line at any point x. Lets look at a few points: f (1) = 32(1) + 96 = 64 Tis says tat te slope of te tangent line at x = 1 is 64, wic confirms wat we i earlier in Example 1. f (3) = 32(3) + 96 = 0 Tis says te slpe of te tangent line at x = 3 is 0, implying tat it is a orizontal line. 2

3 2.1 Oter ways to write te erivative Tese are ifferent ways to write te same ting: f (x) y (x) Example 3: Given y = x, fin y x Solution: x f y x y x + x x x + x x + x ( x + + x) 1 = 1 x + + x 2 x 3 Graping te Derivative Let s look at te grap of f(x) = x ( ) x + + x x + + x Since te erivative is te slope of te tangent lines of f, if f is a line, ten te erivative is simply te slope of te line. Te slope of x wen x < 0 is 1, an 1 wen x > 0. So, we plot tese functions on a grap 3

4 Wen plotting erivatives on our own, we nee to run troug tis list: 1. See were f(x) is flat. Tis tells us te secant line is orizontal, an tus te slope (ence, erivative) is See were f(x) is increasing, or moving up as x gets bigger. Tis says tat te erivative is positive 3. See were f(x) is ecreasing, or moving ownwar as x gets bigger. Tis says tat te erivative is negative. 4 Wen is a Function NOT Differentiable? A function f(x) is ifferentiable at a point x = a wen te erivative exists at tat point. In te example of te function f(x) = x, it is NOT ifferentiable at te point x = 0 because te erivative oes not exist at tat point. Tis is calle a corner, wen a grap as a sarp turn at a point. Anoter grap wen a function is not continuous is calle a cusp. An example is sown ere 4

5 5 Rules of Differentiation All uner te assumption tat f (x) an g (x) exist everywere. 1. Derivative of a constant is 0. If c is a real number ten, 2. Power Rule If n is a real number, ten x xn = nx n 1 3. Constant Multiple x (cf(x)) = cf (x) 4. Sum/Difference x [f(x) ± g(x)] = f (x) ± g (x) 5. x ex = e x x (c) = 0 Example: Derivative of a Polynomial: Let f(x) = 3x 4 +2x 2 +x+1. Ten, using rules 1-3, we ave f (x) = 12x 3 + 4x Higer Orer Derivatives Because f (x) is a ifferentiable function, we can take its erivative. We write tis as x f (x) = f (x) or 2 f x 2 In general, you can take as many erivatives as you want. Te n-t erivative can be written as f (n) (x) Example Given te polynomial above, fin f (x) an f (3) (x) Solution: f (x) = 36x f (3) (x) = 72x 7 Homework Problems 1. Let f(x) = { mx + b x < 2 x 2 x 2 5

6 Fin constants m an b suc tat f(x) is ifferentiable everywere. Solution: For f(x) to be ifferentiable, it must be a) continuous an b) ave a continuous erivative everywere. Let s work on b) first: We can write te erivative of f(x) as { f m x < 2 (x) = 2x x 2 So, for f (x) to be continuous at x = 2, we nee te erivative on te left (m) to equal te erivative on te rigt (2x). Tis gives tat m = 2(2) = 4. So we ave tat f(x) = { 4x + b x < 2 x 2 x 2 For f(x) to be continuous at x = 2, we nee te lim f(x) = f(2). We x 2 know tat f(2) = 2 2 = 4 f(x), so we nee tat x 2 + lim x 2 giving tat b = 4 2. Fin x (10x4 32x + e 2 ) f(x) (4x + b) = 4(2) + b = 4 x 2 Solution: f (x) = 40x 3 32 (Since e 2 is a constant). 3. Fin x (x2 + 1) 2 Solution: After factoring we see f(x) = x 4 + 2x 2 + 1, so f (x) = 4x 3 + 4x 4. Fin x 2 x x 2 Solution Multiplying top an bottom by te conjugate of te enominator gives tat f(x) = x + 2, so f (x) = 1 2 x 6