# 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.