Calculus: Differentiation Formulas

Derivative of a Constant

• The derivative of a constant is always 0.

Power Rule

• Applies to a variable raised to a constant:
  • ( f(x) = x^n ), the derivative ( f'(x) = nx^{n-1} ).
  • Examples:
    • Derivative of ( x^3 ) is ( 3x^2 ).
    • Derivative of ( x^4 ) is ( 4x^3 ).

Exponential Functions

• Derivative of a constant raised to a variable:
  • ( f(x) = a^x ), derivative ( f'(x) = a^x \ln(a) ).
• For a function of ( x ):
  • ( f(x) = a^u ), derivative ( f'(x) = a^u \cdot u' \cdot \ln(a) ).

Logarithmic Differentiation

• Used for variables raised to variables.
• Recommended resource: YouTube video on logarithmic differentiation.

Constant Multiple Rule

• Derivative of a function times a constant:
  • ( f(x) = c \cdot g(x) ), derivative ( f'(x) = c \cdot g'(x) ).
  • Example: Derivative of ( 5x^4 ) is ( 20x^3 ).

Product Rule

• For two functions ( u ) and ( v ):
  • ( (uv)' = u'v + uv' ).

Quotient Rule

• For division of two functions ( u ) and ( v ):
  • ( \left(\frac{u}{v}\right)' = \frac{v u' - u v'}{v^2} ).

Chain Rule

• For composite functions:
  • ( f(g(x)) ): derivative is ( f'(g(x)) \cdot g'(x) ).
  • If ( f(g(u)) ), it's ( f'(g(u)) \cdot g'(u) \cdot u' ).
• Combined with Power Rule:
  • ( (f(x)^n)' = n \cdot f(x)^{n-1} \cdot f'(x) ).

Derivative of Logarithmic Functions

• ( \log_a(u) ): derivative ( \frac{u'}{u \ln(a)} ).
• ( \ln(u) ): derivative ( \frac{u'}{u} ).

Trigonometric Functions

• ( \sin(u) ): derivative ( \cos(u) \cdot u' ).
• ( \cos(u) ): derivative ( -\sin(u) \cdot u' ).
• ( \tan(u) ): derivative ( \sec^2(u) \cdot u' ).
• ( \csc(u), \sec(u), \cot(u) ): derivatives involve their reciprocal identities and chain rule.

Inverse Trigonometric Functions

• ( \sin^{-1}(u) ): ( \frac{u'}{\sqrt{1-u^2}} ).
• ( \cos^{-1}(u) ): ( -\frac{u'}{\sqrt{1-u^2}} ).
• ( \tan^{-1}(u) ): ( \frac{u'}{1+u^2} ).
• ( \cot^{-1}(u) ): ( -\frac{u'}{1+u^2} ).
• ( \sec^{-1}(u) ), ( \csc^{-1}(u) ): derivatives involve complex forms.

Summary

• Essential differentiation formulas for calculus.
• Recommended for students preparing for derivatives tests.
• Additional resources available in description links.