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

27. Find values of ( x ) such that the tangent line to ( y = e^{x} ) is parallel to the line defined by ( 12x - 2y - 6 = 0 ).
28. 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.