Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics General Education Department Mathematics

DERIVATIVES Instantaneous rate of change of a Function is called the limit which is equivalent to that of finding a tangent line to the graph of the function. Definitions: assume that a chemical reaction is taking place and that f(t) designates the amount of a substance present t units of time after the reaction starts. Then the change in the amount of the substance from time t 1 to time t 2 is f(t 2 ) – f(t 1 ), and the average rate of change of the amount per unit of time during the time interval [t 1, t 2 ] is ( f(t2) – f(t1))/(t2 - t1).

DERIVATIVES Definition The derivative of a function f is the function f ’ defined by f’(a) = limit f(x) – f(a) The domain of f ` is the set consisting of every number a at which the above limit exists. If we let h = x – a so that x = h + a then h is close to 0 when x is close to a. Therefore we have: x - a x a f`(a) = limit f(a + h) – f(a) h 0h

EXAMPLES 1. If f(x) = x 2, find the derivative f ` of f. f ’ defined by f’(a) = limit x 2 – a 2 x - a x a f`(a) = limit (x + a) = 2a x a

EXAMPLES 2. If g(x) = 1/x, find the derivative g ` of g. g`(a) = limit g(a + h) –g(a) h 0 h g`(a) = limit 1/(a + h) – 1/a h 0 h g`(a) = limit a – h – a ah(a + h) h 0 = -1/a 2

Skill practice 1. If g(x) = √x, find the derivative g ` of g. 2. If f(x) = x 3 - 1, find the derivative f ` of f.

Differentiation formulas The derivative of f(x) = k where k is a constant is f ` = 0. The derivative of f(x) = x n where n is a positive integer is given by f ` = n x n-1

Examples 1. find the derivative of the function f(x) = 5x 4 – 3x 3 + 6x f ` = 20x 3 – 9x x 2. find equation of the tangent line to the graph of the function f(x) = x 2 at the point P(2,4). f ` (x) = 2x. Therefore f ` (2) = 4. equation of the tangent line is 4 = (y-4)/(x-2) which equals 4x – 8 = y -4 y = 4x - 4

Example Graph of function y = x 2, tangent at P(2,4) x y 0 P(2,4)

Skill practice 1. Find the derivative of the function f(x) = 4x 5 – 2x 4 + 8x 2 + 7x find equation of the tangent line to the graph of the function f(x) = x 2 at the point P(-2,4). 3. Find equation of the tangent line to the graph of the function f(x) = x at the point P(-1,1).

Example Find the derivative of the function f(x) = x 2 + 2x + 4 using a) first principle b)power rule Solution: a) y = x 2 + 2x + 4 a) ∆y + y = (∆x + x) 2 + 2(∆x + x) + 4 ∆y = -y + (∆x) 2 + 2x ∆x + x ∆x + 2x +4 ∆y = -x 2 -2x -4 + (∆x) 2 +2x ∆x +x 2 +2 ∆x +2x+4 ∆y = (∆x) 2 + 2x ∆x + 2 ∆x. Taking limit both sides lim ∆y = (∆x) 2 + 2x ∆x + 2 ∆x = 2x + 2 ∆x 0 ∆x ∆x

Example b) using the power rule: dy/dx = 2x + 2

Skill practice Find the derivative of the function f(x) = x 2 + 3x + 5 using a) first principle b) power rule

The slope of a curve is variable The slope of the curve along any interval is given approximately by the slope of the chord along that same interval. The slope of the curve is changing as the slope of each chord is different. A B C D

Slope of y=x 2 using first principle y=x 2 means y + ∆y = (x + ∆x) 2 then ∆y = -y + x 2 +2x ∆x + (∆x) 2 means ∆y =- x 2 +x 2 + 2x ∆x+(∆x) 2 lim ∆y = (∆x) 2 + 2x ∆x then dy/dx = 2x ∆x 0 ∆x ∆x A B A B B*(∆x,∆y)

Notations for derivatives Notations used for derivative of y=f(x): dy/dx df(x)/dx df/dx f ` (x) y`y` Dy Df(x)

PRODUCT AND QUOTIENT DIFFERENTIATION FORMULAS If f and g are two functions then D(fg) = f Dg + g Df If f and g are two functions then D(f/g) = (g Df – f Dg)/ g 2

Examples 1. Find the derivative of F(x) = x 2 /(x 2 – 4) DF(x) = ((x 2 – 4) 2x - x 2 2x)/ (x 2 – 4) 2 ) = -8x/ (x 2 – 4) 2 2. Find the derivative of F(x) = (x 3 +2x) (x 2 – 4) DF(x) = (x 3 +2x) 2x + (x 2 – 4) (3x 2 +2)

Skill practice 1. Find the derivative of F(x) = x 3 /(x 3 – 4) 2. Find the derivative of F(x) = (x 4 +3x) (x 2 – 5)

Slope of a curve and turning point The Slope of the curve at a point is the same as the slope of the tangent at that point. a b c d at turning point a, b, c and d slope dy/dx = 0. Local min c Local max b Absolute min a Absolute max d

Turning points To find the x coordinates of the turning points for y=f(x), the following method is used. Step 1: Find dy/dx for the given curve y=f(x). Step2: Solve the equation dy/dx = 0.

Examples Find the turning point for the following functions: 1. y = 3x 2 -18x y = x 3 -1 Solution: dy/dx = 6x -18 when dy/dx = 0 we have 0 = 6x -18 which is x = 3. when x = 3 then y=7. the turning point exists at x=3 and y=7. 2. dy/dx = 3x 2, when dy/dx = 0 we have 0 = 3x 2 means x =0. when x=0 then y=-1. The turning point exists at x=0 and y = -1

Skill practice Locate the turning point for the following functions: 1. y = x 2 - 6x y = 2x 3 - 3x 2 3. y = x 4 – 2 x 2 4. y = x 3 - 3x 2 - 9x

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