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 2 + 96x. 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 2 + 96x) 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 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 2 + 96x. Solution: f (x) f(x + ) f(x) f(x) f(x+) { }} { { }} { 16(x + ) 2 + 96(x + ) ( 16x 2 + 96x) 16(x 2 + 2x + 2 ) + 96x + 96 + 16x 2 96x ( 32x + 96 16) ( 32x + 96 16) = 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
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
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 0. 2. 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 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 + 1 6 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 2 + 4 f (3) (x) = 72x 7 Homework Problems 1. Let f(x) = { mx + b x < 2 x 2 x 2 5
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