Derivatives  Definition of a Derivative  Power Rule  Package Rule  Product Rule  Quotient Rule  Exponential Function and Logs  Trigonometric Functions Barbara Wong Period 5

Definition of a Derivative  The slope of the tangent line to the graph of a function at a given point.  Mathematical Formula: f ’ (x) = Lim f (x+h) – f (x) h0 h

Power Rule  y=  x  y’=   x -1  Example: f(x) = 8x 3 f '(x) = 24x 2

Package Rule  d[a() n ] = na   n-1  d dx dx  Example: f(x) = 2(x 2 -1) 2 f '(x) = 4(x 2 -1) 1  (x 2 -1)’ = 8x(x 2 -1)

Product Rule  d (  ) =   d  +   d dx dx dx  ’  +   ’  Example: f(x) = 3x e x f '(x) = 3e x + 3xe x = 3e x (x + 1)

Quotient Rule  d(/  ) =   ddx    d  dx dx  2  ’  -   ’  2  Example: f (x) = x x 3 f’(x) = (2x  x 3 ) – (x 2 + 1)  3x 2 x 6 = 2 x 4 – 3x 4 – 3x 2 = – x 4 – 3x 2 x 6 x 6 = – x 2 – 3 x 6

Rules for Simplifying Logs 1.ln(  ) = ln() + ln(  ) 2.ln  = ln() – ln(  )  3.ln  =  ln() Examples: 1.ln(2x) = ln(2) + ln(x) 2.ln(x/2)= ln(x) – ln(2) 3.lnx 2 = 2lnx

Rules for Simplifying Natural Logs and Exponentials 1.ln(e  ) =  2.e ln =  (e x and lnx are inverse functions) Examples: 1. ln(e 2 ) = 2 2. e ln2 = 2

Derivatives of the Logarithm & Exponential Functions 1.f(x) = ln f’(x) = 1 (d)   Example: f(x) = ln x f '(x) = 1/x 2.f(x) = e  f’(x) = e  (d)  Example: f(x) = e 3x f '(x) = e 3x  3 = 3e 3x

Derivatives of Trigonometric Functions  d(sin) dx = cos  d /dx  d(cos) dx = -sin  d /dx  d(tan) dx = sec 2   d /dx  d(cot) dx = -csc 2   d /dx  d(sec) dx = sec  tan  d/dx  d(csc) dx = -csc  cot  d/dx