MCV4U – Grade 12 Calculus & Vectors – Sinusoidal and Logarithmic Derivatives

Grade 12 – Calculus and Vectors

 

Sinusoidal and Logarithmic Derivatives

 

Derivatives Sine and Cosine functions

  • Derivative of y = sin x is y’ = cos x
  • Derivative of y = cos x is y’ = -sin x
  • Derivative of y = sin (f(x)) is y’ = cos (f(x)) * f’(x)
  • Derivative of y = cos (f(x)) is y’ = -sinx (f(x)) * f’(x)
  • Derivative of y = sin2 (f(x)) is y’ = 2 cos (f(x)) * f’(x)
  • Derivative of y = cos2 (f(x)) is y’ = -2 sin (f(x)) * f’(x)

 

Simple Harmonic Motion Trigonometric Application Problems

  • 1st Derivative used to find velocities
  • 2nd Derivatives used to find accelerations as well as max/min velocities
  • To find period from its equation, it’s 360/k or 2 Pi / k for radians

 

The number e

  • The symbol e is defined as limit when n -> infinity (1 + 1/n)n . The value is ~ 2.71
  • Rate of change of exponential function is also exponential
  • Derivative of y = ex is y = ex (the same as original function)

 

Natural Logarithm

  • Lnx = logex
  • The functions y = lnx and y = ex are inverses

 

Derivatives of exponential functions

  • The derivative of y = bx is y’ = bx * lnx
  • The derivative of y = bf(x) is y’ = bf(x) * ln b * f’(x)
  • The derivative of y = ef(x) is y’ = ef(x) * f’(x)
  • You solve most logarithms by applying ln or log both sides and isolating the variable.