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Rules for Derivatives of Trigonometric Functions

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  • Derivative of \(\sin x\)
    The derivative of \(\sin x\) is \(\cos x\).
  • Derivative of \(\cos x\)
    The derivative of \(\cos x\) is \(-\sin x\).
  • Derivative of \(\tan x\)
    The derivative of \(\tan x\) is \(\sec^2 x\).
  • Derivative of \(\cot x\)
    The derivative of \(\cot x\) is \(-\csc^2 x\).
  • Derivative of \(\sec x\)
    The derivative of \(\sec x\) is \(\sec x \tan x\).
  • Derivative of \(\csc x\)
    The derivative of \(\csc x\) is \(-\csc x \cot x\).
  • General rule for derivative of trig functions
    Use the standard derivatives of sine, cosine, tangent, cotangent, secant, and cosecant combined with the chain rule if the argument is not \(x\).
  • Derivative of \(\sin(kx)\) where \(k\) is constant
    The derivative is \(k \cos(kx)\).
  • Derivative of \(\cos(kx)\) where \(k\) is constant
    The derivative is \(-k \sin(kx)\).
  • Why is the derivative of \(\tan x\) equal to \(\sec^2 x\)?
    Because \(\tan x = \frac{\sin x}{\cos x}\), applying the quotient rule yields \(\sec^2 x\).
  • Derivative of \(\cot x\) using quotient rule
    Since \(\cot x = \frac{\cos x}{\sin x}\), its derivative is \(-\csc^2 x\).
  • Relationship between derivatives of \(\sec x\) and \(\tan x\)
    The derivative of \(\sec x\) involves \(\tan x\) as \(\sec x \tan x\).
  • Derivative of \(\csc x\) involves which trig functions?
    It involves \(\csc x\) and \(\cot x\) as \(-\csc x \cot x\).
  • How to remember the sign of trig derivatives?
    Derivatives of sine and cosecant are positive times cosine and cotangent respectively; cosine, cotangent, and cosecant derivatives have a negative sign.
  • Derivative of \(\sin(-x)\)
    Using chain rule, derivative is \(-\cos(-x) = -\cos x\).
  • Derivative of \(\cos(-x)\)
    Using chain rule, derivative is \(\sin(-x) = -\sin x\).
  • Derivative of \(\tan(kx + c)\)
    The derivative is \(k \sec^2(kx + c)\) where \(k, c\) are constants.
  • Use of chain rule with trig derivatives
    Multiply the derivative of the trig function by the derivative of its inner function.
  • Derivative of \(\sec^2 x\)
    Use the chain rule: derivative is \(2 \sec x \sec x \tan x = 2 \sec^2 x \tan x\).
  • Derivative of \(\csc^2 x\)
    Using chain rule, derivative is \(2 \csc x (-\csc x \cot x) = -2 \csc^2 x \cot x\).