Derivatives of Exponential and Logarithmic Functions

Exponential and Logarithmic Functions

Introduction to Exponential and Logarithmic Differentiation

Exponential and logarithmic functions are fundamental in calculus, especially in business applications involving growth and decay, interest calculations, and elasticity. Their derivatives are essential for solving optimization and modeling problems.

Differentiation of Logarithmic Functions

Derivative of the Natural Logarithm Function

The derivative of the natural logarithm function, ln(x), is a foundational result in calculus. It is used extensively in simplifying the differentiation of more complex functions.

• Key Formula: The derivative of $\ln(x)$ with respect to $x$ is: $$ \frac{d}{dx} \ln(x) = \frac{1}{x}$$

• Chain Rule Application: For a composite function $\ln(u(x))$: $$ \frac{d}{dx} \ln(u(x)) = \frac{1}{u(x)} \cdot u'(x)$$

• Example: If $y = \ln(3x^2)$, then: $$ \frac{dy}{dx} = \frac{1}{3x^2} \cdot 6x = \frac{2}{x}$$

Differentiation of Exponential Functions

Derivative of the Exponential Function

The exponential function $e^{x}$ and its generalizations are widely used in business calculus for modeling continuous growth and decay. Their derivatives are straightforward but require careful application of the chain rule for composite exponents.

• Key Formula: The derivative of $e^{x}$ is: $$ \frac{d}{dx} e^{x} = e^{x}$$

• Chain Rule Application: For $e^{u(x)}$: $$ \frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)$$

• Example: If $f(x) = 5e^{3x}$, then: $$ f'(x) = 5 \cdot e^{3x} \cdot 3 = 15e^{3x}$$

Product Rule with Exponential Functions

Differentiating Products Involving Exponentials

When differentiating a product of a polynomial and an exponential function, the Product Rule is used. This is common in business calculus when modeling revenue, cost, or population functions.

• Product Rule: For $f(x) = u(x) v(x)$: $$ f'(x) = u'(x) v(x) + u(x) v'(x)$$

• Example: Differentiate $f(x) = x^5 e^{5x}$:

  • Let $u(x) = x^5$, $v(x) = e^{5x}$

  • $u'(x) = 5x^4$

  • $v'(x) = e^{5x} \cdot 5$

  • So, $$ f'(x) = 5x^4 e^{5x} + x^5 \cdot 5e^{5x} = (5x^4 + 5x^5) e^{5x}$$

Inverse Relationship of Exponential and Logarithmic Functions

Understanding Inverse Functions

The exponential function $e^x$ and the natural logarithm $\ln(x)$ are inverse functions. This relationship is crucial for solving equations and understanding their derivatives.

• Inverse Property: $e^{\ln(x)} = x$ and $\ln(e^x) = x$ for $x > 0$.

• Application: This property is often used to simplify expressions before differentiation or integration.

Summary Table: Key Derivative Rules

Function	Derivative
$\ln(x)$	$\frac{1}{x}$
$e^{x}$	$e^{x}$
$e^{u(x)}$	$e^{u(x)} \cdot u'(x)$
$\ln(u(x))$	$\frac{1}{u(x)} \cdot u'(x)$

Additional info: These rules are foundational for business calculus, especially in applications involving continuous growth, decay, and elasticity analysis.