By: Jane Kim, Period 4, 2007 Source: Mr. Wiencek’s Noted from Class

The Definition of Derivative f(x) (x,0) (x,f(x)) ((x+h),f(x+h)) h ((x+h),0)

Limit Definition of a Derivative h  0 lim f ‘(x) = f(x+h) – f(x) h f ‘(x) = lim h  0 f(x+h) – f(x) h

Remember! Don’t forget to write out the Limit Definition of Derivative Remember to write everytime lim h  0

Example f(x) = 5x + 3 F(x+h) = 5(x+h) + 3 = 5x + 5h + 3 5x + 5h + 3 – (5x + 3) h

Example continued 5x + 5h + 3 – 5x – 3 h 5h h = 5 Answer: 5

Power Rule, Slopes of Tangent Lines f ‘(x)F Prime of x y’ y prime dy dx d dx dy dx Derivatives with respect to x

Common Powers x = x 5 1 x = x 1 x 3 LN Y x = x x = x 4 3 x x x x x x 1/2 1/ /3

Power Rule y = x y’ = x * – 1

Example y = 3x - x + 2 y’ = 2(3)x – 1(-1) + 0(2) = 6x + (-1)x + 0x = 6x – Answer: = 6x – 1 dy dx dy dx

Remember! Derivatives = Slope

Example y = 2x x = 0,1,3,-4 f(0) = 4(0) = 0 f(1) = 4(1) = 4 f(3) = 4(3) = 12 f(-4) = 4(-4) = -16 dy dx = 4x 2

Graphs & Using the Derivative to find Slope Tangent Line Slope = m Normal Line Slope = 1 m

Example y = 2x + 3 Find the equation of a)The tangent at 1 b)The normal at 1 y = 2x + 3 y’= 6x + 3 y(1) = 6(1) + 3 = = (1, ?) **Derivatives = Slope Slope = 9

Example continued To find y: plug x = 1 back into the original equation, y = 2x + 3 y = 2(1) + 3(1) = = 5 so (1,5) 3 3 (1, ?)

Example continued Tangent equation: y – y = m(x – x ) y – 5 = 9(x – 1) y – 5 = 9x – 9 y = 9x – 4 Normal equation: y – y = - 1/9(x – x ) y – 5 = - 1/9(x – 1) y – 5 = -1/9x + 1/9 y = -1/9x + 46/

THE END