Exponential and logarithmic functions are key players in calculus. They model growth, decay, and many real-world phenomena. Their unique properties make them essential for solving complex problems and understanding natural processes.

Differentiating these functions requires special techniques. The derivatives of exponential functions involve the original function, while logarithmic derivatives involve reciprocals. These rules are crucial for tackling advanced calculus problems and real-world applications.

Exponential and Logarithmic Differentiation

Differentiation of exponential functions

• The derivative of an exponential function $a^x$ is $a^x \ln(a)$, where $a$ is a positive constant not equal to 1
  • Multiplies the original function by the natural logarithm of the base
  • Example: $\frac{d}{dx}2^x = 2^x \ln(2)$
• The derivative of the natural exponential function $e^x$ is itself, $e^x$
  • Makes $e^x$ a unique and important function in calculus
  • Example: $\frac{d}{dx}e^x = e^x$
• When an exponential function is multiplied by a constant $k$, the derivative is the constant multiplied by the derivative of the exponential function
  • $\frac{d}{dx}k \cdot a^x = k \cdot a^x \ln(a)$
  • $\frac{d}{dx}k \cdot e^x = k \cdot e^x$
  • Example: $\frac{d}{dx}5e^x = 5e^x$
• For exponential functions with a non-trivial argument $u(x)$, the chain rule is applied
  • $\frac{d}{dx}a^{u(x)} = a^{u(x)} \cdot \ln(a) \cdot u'(x)$
  • $\frac{d}{dx}e^{u(x)} = e^{u(x)} \cdot u'(x)$
  • Example: $\frac{d}{dx}e^{\sin x} = e^{\sin x} \cdot \cos x$
• Exponential functions are often used to model exponential growth and decay in various applications

Derivatives of logarithmic functions

• The derivative of a logarithmic function $\log_a(x)$ is $\frac{1}{x \ln(a)}$, where $a$ is a positive constant not equal to 1 and $x$ is positive
  • Divides 1 by the product of $x$ and the natural logarithm of the base
  • Example: $\frac{d}{dx}\log_2(x) = \frac{1}{x \ln(2)}$
• The derivative of the natural logarithm $\ln(x)$ is $\frac{1}{x}$, where $x$ is positive
  • Simplifies the derivative compared to other logarithmic functions
  • Example: $\frac{d}{dx}\ln(x) = \frac{1}{x}$
• When a logarithmic function is multiplied by a constant $k$, the derivative is the constant divided by the product of $x$ and the natural logarithm of the base
  • $\frac{d}{dx}k \cdot \log_a(x) = \frac{k}{x \ln(a)}$
  • $\frac{d}{dx}k \cdot \ln(x) = \frac{k}{x}$
  • Example: $\frac{d}{dx}3\ln(x) = \frac{3}{x}$
• For logarithmic functions with a non-trivial argument $u(x)$, the chain rule is applied
  • $\frac{d}{dx}\log_a(u(x)) = \frac{u'(x)}{u(x) \ln(a)}$
  • $\frac{d}{dx}\ln(u(x)) = \frac{u'(x)}{u(x)}$
  • Example: $\frac{d}{dx}\ln(\cos x) = -\tan x$
• Understanding logarithmic properties is crucial for simplifying and differentiating logarithmic expressions

Logarithmic differentiation techniques

• Logarithmic differentiation is a technique used to differentiate functions involving products, quotients, or complex exponents by taking the natural logarithm of both sides of the equation
  • Simplifies the equation using logarithm properties ($\ln(ab) = \ln(a) + \ln(b)$, $\ln(a^b) = b\ln(a)$)
  • Differentiates both sides using implicit differentiation
  • Solves for the derivative of the original function
• Logarithmic differentiation is particularly useful for functions of the form $f(x) = [u(x)]^{v(x)}$ or $f(x) = u_1(x) \cdot u_2(x) \cdot ... \cdot u_n(x)$
  • Converts products to sums and exponents to coefficients
  • Example: $f(x) = x^x$, $f(x) = x\sqrt{x+1}$
• Steps for logarithmic differentiation:
  1. Take the natural logarithm of both sides of the equation
  2. Use logarithm properties to simplify the equation
  3. Differentiate both sides of the simplified equation using implicit differentiation
  4. Solve for the derivative of the original function
• Example: To find $\frac{d}{dx}x^x$ using logarithmic differentiation:
  1. $\ln(y) = \ln(x^x)$
  2. $\ln(y) = x \ln(x)$
  3. $\frac{1}{y} \cdot \frac{dy}{dx} = \ln(x) + 1$
  4. $\frac{dy}{dx} = x^x (\ln(x) + 1)$

Advanced Techniques and Applications

• The inverse function rule is useful when differentiating inverse functions, including logarithmic and exponential functions
• L'Hôpital's rule is a powerful technique for evaluating limits involving indeterminate forms, often used with exponential and logarithmic functions
• These advanced techniques are particularly useful in solving complex problems involving exponential and logarithmic functions