MA129: Week 6 Study Notes

Course Information

  • Course: MA129
  • Week: 6
  • Instructor: Dr. L. Howe
  • Department: Mathematics, Wilfrid Laurier University

Differentiation as a Process

Notation for Derivatives

  • Let ( y = f(x) )
    • The derivative is represented as:
    • ( f'(x) ) or ( y' ) (Easiest notation)
    • ( \frac{dy}{dx} ) or ( \frac{df}{dx} ) (Leibniz notation: useful for later applications)
    • For a small change in ( x ) denoted by ( h ):
    • ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} )
    • More explicitly, ( f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} )
    • The derivative can also be denoted as ( \frac{d}{dx}[f(x)] ) (useful for formulas)

Example of Notation

  • Example: Let ( y = f(x) = x^2 ). Its derivative is ( 2x ). This is correct in any of the following forms:
    • ( f'(x) = 2x )
    • ( y' = 2x )
    • ( \frac{dy}{dx} = 2x )
    • ( \frac{df}{dx} = 2x )
    • ( \frac{d}{dx}[x^2] = 2x )

Constant Rule

  • If ( f(x) = k ) where ( k ) is a constant, then:
    • ( f'(x) = 0 )
    • Or, in Leibniz notation: ( \frac{d}{dx}[k] = 0 )

Power Rule

  • If ( f(x) = x^n ), where ( n ) is any real number, then:
    • ( f'(x) = nx^{n-1} )
    • In Leibniz notation: ( \frac{d}{dx}[x^n] = nx^{n-1} )
    • Mental Instructions: Bring the exponent down in front as a multiplier and decrease the exponent by 1.

Examples

  1. If ( f(x) = e ): \
    • ( f'(x) = 0 ) since ( f(x) ) is constant.
  2. If ( f(x) = x^2 ): \
    • Using the Power Rule, ( f'(x) = 2x^{1} = 2x ).
  3. ( \frac{d}{dx}[x] = \frac{d}{dx}[x^{1}] = 1 \cdot x^{0} = 1 ).
  4. If ( f(x) = \frac{1}{x^2} = x^{-2} ): \
    • ( f'(x) = -2x^{-3} ).
  5. For ( g(t) = \sqrt[3]{t} ): find ( g'(8) ):
    • ( g(t) = t^{1/3} )
    • ( g'(t) = \frac{1}{3}t^{-2/3} )
    • ( g'(8) = \frac{1}{3} 8^{-2/3} = \frac{1}{12} )

Constant Multiple Rule

  • Let ( k ) be a real number. If ( f(x) ) is differentiable, then so is ( kf(x) ), and:
    • ( \frac{d}{dx}[kf(x)] = k \frac{d}{dx}[f(x)] )
    • Example: If ( f(x) = 5x^2 ): \
    • ( \frac{d}{dx}[5x^2] = 5 \frac{d}{dx}[x^2] = 5 \cdot 2x = 10x ) \
    • Therefore, ( f'(x) = 10x )

Sum & Difference Rules

  • If both functions ( f(x) ) and ( g(x) ) are differentiable, then:
    • ( \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)] )
    • ( \frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}[f(x)] - \frac{d}{dx}[g(x)] )

Product and Quotient Rules

Product Rule

  • If both functions ( f(x) ) and ( g(x) ) are differentiable, then:
    • ( \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) )
    • Mental Instructions: The derivative of the first function, times the second function, plus the derivative of the second function, times the first.

Quotient Rule

  • If both functions ( f(x) ) and ( g(x) ) are differentiable, then:
    • ( \frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^{2}} )
    • Mental Instructions: The derivative of the top function times the bottom function minus the derivative of the bottom function times the top function, all over the bottom function squared.

Chain Rule

  • If two functions ( f(x) ) and ( g(x) ) are differentiable, then:
    • ( \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) )
    • Mental Instructions: The derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Derivative of the Exponential Function

  • The base ( e ) is commonly used because:
    • ( \frac{d}{dx}[e^x] = e^x )
    • Using this and the Chain Rule to compute ( \frac{d}{dx}[a^x] ) for any ( a > 0 ):
      • ( a^x = e^{x ext{ln} a} )
    • Therefore, ( \frac{d}{dx}[a^x] = a^x ext{ln} a )

Derivatives of Logarithmic Functions

  • The derivative of the natural logarithm is given by:
    • ( \frac{d}{dx}[ ext{ln} x] = \frac{1}{x} ) (for ( x > 0 ))
    • Using the Change of Base Formula:
    • ( ext{log}_a x = \frac{ ext{ln} x}{ ext{ln} a} ) gives:
    • ( \frac{d}{dx}[ ext{log}_a x] = \frac{1}{x ext{ln} a} )

End of Notes

  • This document contains a comprehensive overview of differentiation rules, procedures, and examples essential for understanding calculus principles related to derivatives.