4.3 Derivatives of Exponential and Logarithmic Functions

Exponential and logarithmic functions are key players in calculus. They have unique properties that make them essential for modeling growth, decay, and other real-world phenomena. Their derivatives follow special rules that simplify many calculations.

Understanding these functions and their derivatives is crucial for tackling more complex problems. They're used in everything from compound interest to population growth, making them indispensable tools in your calculus toolkit.

Exponential Functions

Natural Exponential Function and Euler's Number

Top images from around the web for Natural Exponential Function and Euler's Number

• 

  Exponential function - Wikipedia View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: The Natural Exponential Function View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: The Natural Exponential Function View original

  Is this image relevant?

• 

  Exponential function - Wikipedia View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: The Natural Exponential Function View original

  Is this image relevant?

1 of 3

Top images from around the web for Natural Exponential Function and Euler's Number

• 

  Exponential function - Wikipedia View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: The Natural Exponential Function View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: The Natural Exponential Function View original

  Is this image relevant?

• 

  Exponential function - Wikipedia View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: The Natural Exponential Function View original

  Is this image relevant?

1 of 3

• The natural exponential function is defined as $f(x) = e^x$
• $e$ is Euler's number, an important mathematical constant approximately equal to 2.71828
• Euler's number ($e$) is defined as the limit of $(1 + \frac{1}{n})^n$ as $n$ approaches infinity
• The natural exponential function has the unique property that its derivative is itself: $\frac{d}{dx}e^x = e^x$
• This property makes the natural exponential function fundamental in calculus and mathematical analysis

Exponential Function with Base $a$

• The exponential function with base $a$ is defined as $f(x) = a^x$, where $a$ is a positive real number not equal to 1
• The derivative of the exponential function with base $a$ is given by $\frac{d}{dx}a^x = a^x \ln(a)$
• The exponential function with base $a$ can be expressed in terms of the natural exponential function: $a^x = e^{x\ln(a)}$
• This relationship allows for the calculation of derivatives and integrals of exponential functions with any base using the properties of the natural exponential function
• Common bases for exponential functions include 2 (binary exponential function) and 10 (decimal exponential function)

Logarithmic Functions

Natural Logarithm and Logarithmic Function with Base $a$

• The natural logarithm, denoted as $\ln(x)$ or $\log_e(x)$, is the inverse function of the natural exponential function
• The natural logarithm is defined for all positive real numbers $x$ and satisfies the equation $\ln(e^x) = x$
• The derivative of the natural logarithm is given by $\frac{d}{dx}\ln(x) = \frac{1}{x}$
• The logarithmic function with base $a$, denoted as $\log_a(x)$, is the inverse function of the exponential function with base $a$
• The logarithmic function with base $a$ is related to the natural logarithm by the change of base formula: $\log_a(x) = \frac{\ln(x)}{\ln(a)}$
• The derivative of the logarithmic function with base $a$ is given by $\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$

Logarithmic Differentiation

• Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of simpler functions
• The process involves taking the natural logarithm of both sides of an equation and then differentiating using the properties of logarithms
• Logarithmic differentiation is particularly useful when dealing with functions of the form $f(x) = [g(x)]^{h(x)}$
• The steps for logarithmic differentiation are:
  1. Take the natural logarithm of both sides of the equation
  2. Use the properties of logarithms to simplify the equation
  3. Differentiate both sides of the equation using the chain rule
  4. Solve for the derivative of the original function
• Logarithmic differentiation simplifies the process of finding derivatives for complex functions, such as $f(x) = x^x$ or $f(x) = (\sin(x))^{\cos(x)}$