Lecture on Finding Derivatives

Derivative of a Constant

• The derivative of any constant is always 0.
  • Example: Derivative of 7, -4, π, or π^e is 0.

Power Rule

• For a variable raised to a constant power ( x^n ), the derivative is ( n \times x^{n-1} ).
  • Example: Derivative of ( x^2 ) is ( 2x ).
  • Example: Derivative of ( x^3 ) is ( 3x^2 ).

Constant Multiple Rule

• The derivative of a constant times a function is the constant times the derivative of the function.
  • Example: ( 4x^5 ) derivative is ( 20x^4 ).

Polynomial Functions

• Derivative of a polynomial is the derivative of each term.
  • Example: ( f(x) = 4x^3 + 7x^2 - 9x + 5 )
    • ( f'(x) = 12x^2 + 14x - 9 ).

Rational Functions

• Use the power rule with negative exponents.
  • Example: Derivative of ( \frac{1}{x^2} ) becomes ( -\frac{2}{x^3} ).

Radical Functions

• Rewrite radicals as exponents.
  • Example: Derivative of ( \sqrt{x} ) is ( \frac{1}{2\sqrt{x}} ).

Trigonometric Functions

• Sine: ( \frac{d}{dx}[\sin u] = \cos u \cdot u' )
• Cosine: ( \frac{d}{dx}[\cos u] = -\sin u \cdot u' )
• Tangent: ( \sec^2 u \cdot u' )
• Patterns:
  • Derivatives of functions starting with 'c' have negative signs.

Natural Logs

• ( \frac{d}{dx}[\ln u] = \frac{u'}{u} )
  • Example: ( \ln x ) is ( \frac{1}{x} ).

Logarithmic Functions

• ( \frac{d}{dx} [\log_a u] = \frac{u'}{u \ln a} )

Exponential Functions

• ( \frac{d}{dx}[a^u] = a^u \cdot u' \cdot \ln a )
• Special case for ( e^x ): ( \frac{d}{dx}[e^x] = e^x )

Product Rule

• ( \frac{d}{dx}[u \cdot v] = u'v + uv' )
  • Example: ( \frac{d}{dx}[x^2 \ln x] = 2x \ln x + x ).

Quotient Rule

• ( \frac{d}{dx}[\frac{u}{v}] = \frac{vu' - uv'}{v^2} )
  • Example: ( \frac{d}{dx}[\frac{3x-5}{7x+4}] ).

Chain Rule

• Differentiate composite functions: ( f(g(x)) )
  • ( f'(g(x)) \cdot g'(x) )
  • Example: Derivative of ( \sin(x^5) ).

Implicit Differentiation

• Differentiate equations involving both x and y.
  • Example: Solve for ( \frac{dy}{dx} ) in ( x^3 + 4xy + y^2 = 9 ).

Logarithmic Differentiation

• For expressions like ( x^x ), use logarithms to simplify before differentiating.
  • ( y = x^x ) leads to ( \ln y = x \ln x ).

Conclusion

• Covered the basics and techniques for finding derivatives of various types of functions.