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Derivatives

Barbara Wong Period 5. Derivatives. Definition of a Derivative Power Rule Package Rule Product Rule Quotient Rule Exponential Function and Logs Trigonometric Functions. Definition of a Derivative. The slope of the tangent line to the graph of a function at a given point.

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Derivatives

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

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

  3. Power Rule • y= x y’=  x-1 • Example: f(x) = 8x3 f '(x) = 24x2

  4. Package Rule • d[a()n] = na n-1d dx dx • Example: f(x) = 2(x2-1)2 f '(x) = 4(x2-1)1 (x2-1)’ = 8x(x2-1)

  5. Product Rule • d ( )= d+ d dx dx dx • ’ + ’ • Example: f(x) = 3x ex f '(x) = 3ex + 3xex = 3ex (x + 1)

  6. Quotient Rule • d(/ )=   ddx    ddx dx 2 • ’ - ’ 2 • Example: f (x) =x2 + 1 x3f’(x) =(2x  x3 ) – (x2 + 1)  3x2 x6 =2 x4 – 3x4 – 3x2 =– x4 – 3x2 x6 x6 =– x2 – 3 x6

  7. Rules for Simplifying Logs • ln() = ln() + ln() • ln  = ln() – ln()  • ln = ln() Examples: • ln(2x) = ln(2) + ln(x) • ln(x/2)= ln(x) – ln(2) • lnx2 = 2lnx

  8. Rules for Simplifying Natural Logs and Exponentials • ln(e) =  • e ln =  (ex and lnx are inverse functions) Examples: • ln(e2) = 2 • eln2 = 2

  9. Derivatives of the Logarithm & Exponential Functions • f(x) = ln f’(x) = 1 (d)  • Example: f(x) = ln x f '(x) = 1/x • f(x) = e f’(x) = e(d) • Example: f(x) = e3x f '(x) = e3x 3 = 3e3x

  10. Derivatives of Trigonometric Functions • d(sin) dx = cosd/dx • d(cos) dx = -sin d/dx • d(tan) dx = sec2 d/dx  • d(cot) dx = -csc2 d/dx • d(sec) dx = sec tan d/dx  • d(csc) dx = -csccot d/dx

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