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Exponential and Logarithmic Derivatives

Derivatives of Exponential and Logarithmic Functions

Basic Derivatives

  • Exponential Functions:

    • If ( f(x) = e^{x} ), then ( f'(x) = e^{x} ).
    • If ( f(x) = e^{g(x)} ), then ( f'(x) = e^{g(x)} g'(x) ).

  • Logarithmic Functions:

    • If ( f(x) = ext{ln}(x) ), then ( f'(x) = \frac{1}{x} ).
    • If ( f(x) = ext{ln}(g(x)) ), then ( f'(x) = \frac{g'(x)}{g(x)} ).

Examples of Derivatives

  1. Function: ( f(x) = e^{x} )

    • Derivative: ( f'(x) = e^{x} )

  2. Function: ( f(x) = e^{x^2} )

    • Derivative: ( f'(x) = e^{x^2} \cdot 2x = 2xe^{x^2} )

  3. Function: ( f(x) = ext{ln}(2x) )

    • Derivative: ( f'(x) = \frac{2}{2x} = \frac{1}{x} )

  4. Function: ( f(x) = ext{ln}(x^5) )

    • Derivative: ( f'(x) = \frac{5}{x} )

  5. Function: ( f(x) = e^{x^3} )

    • Derivative: ( f'(x) = e^{x^3} \cdot 3x^2 = 3x^2 e^{x^3} )

  6. Function: ( f(x) = ext{ln}(3x) )

    • Derivative: ( f'(x) = \frac{3}{3x} = \frac{1}{x} )

Tangent Lines and Rates of Change

  • Finding Tangent Lines:

    • If( f(x) = e^{g(x)} ) and you want the equation of the tangent line at ( x = a ):
    • Find the slope: ( f'(a) )
    • Use point-slope form: ( y - f(a) = f'(a)(x - a) )

  • Equality of Average and Instantaneous Rates:

    • The average rate of change on an interval ( [a, b] ) is given by ( \frac{f(b) - f(a)}{b - a} ).
    • According to the Mean Value Theorem, there exists a point ( c ) in ( (a, b) ) such that the instantaneous rate of change at ( c ) equals the average rate of change.

Example Problems

  1. Find values of ( x ) such that the tangent line to ( y = e^{x} ) is parallel to the line defined by ( 12x - 2y - 6 = 0 ).
  2. Find values of ( x ) such that the tangent line to ( y = x^2 ) is parallel to the line defined by ( 5x + 2y - 109 = 0 ).

Position, Velocity, and Acceleration

  • To find when a particle is at rest or for acceleration:
    • Position function: ( x(t) = e^{t} + e^{-t} )
    • Velocity function: ( v(t) = x'(t) )
    • Acceleration function: ( a(t) = v'(t) )

Conclusion

  • Mastery of exponential and logarithmic derivatives is crucial for solving calculus problems involving slopes, tangent lines, and rates of change.
  • Practice various forms of functions to become proficient in their derivatives.