Fundamental and Derivative Rules

Fundamental Derivatives

• Derivatives of Common Functions:

  • Sine:

  • ( f(x) = \sin x )

  • ( f'(x) = \cos x )

  • Cosine:

  • ( f(x) = \cos x )

  • ( f'(x) = -\sin x )

  • Tangent:

  • ( f(x) = \tan x )

  • ( f'(x) = \sec^2 x )

  • Secant:

  • ( f(x) = \sec x )

  • ( f'(x) = \sec x \tan x )

Power Rule Derivatives

• Power Rule:
  • General formula:
  • For ( f(x) = x^n ), the derivative is ( f'(x) = nx^{n-1} )
  • Example cases:
  • For ( n = \frac{1}{2} ):
    • ( f(x) = x^{1/2} ) ( \Rightarrow f'(x) = \frac{1}{2}x^{-1/2} )
  • For ( n = -1 ):
    • ( f(x) = x^{-1} ) ( \Rightarrow f'(x) = -x^{-2} )
  • For ( n = -2 ):
    • ( f(x) = x^{-2} ) ( \Rightarrow f'(x) = -2x^{-3} )

Natural Logarithm Derivatives

• Natural Logarithm:
  • ( f(x) = \ln x )
  • Derivative: ( f'(x) = \frac{1}{x} )

Exponential Functions Derivatives

• Exponential Function:
  • General form ( f(x) = e^x ) has a unique property:
  • ( f'(x) = e^x )
  • For bases other than 'e' (e.g., ( f(x) = b^x )):
  • Derivative: ( f'(x) = b^x \ln b )

Trigonometric Function Derivatives (with multiple angles)

• Multiple Angle Sine Function:

  • ( f(x) = \sin(mx) )
  • Derivative: ( f'(x) = m\cos(mx) )

• General Summary of Derivatives:

  • Each of these rules helps in calculus for finding the slope or rate of change of the function with respect to the variable x.
  • Mastering these derivatives allows for easier handling of dynamic systems and functions in various fields such as physics, engineering, and economics.