Advanced Mathematics

Calculus

# Differentiating Exponential and Natural Logarithmic Functions

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Differentiating Exponential and Natural Logarithmic Functions

Basics of Differentiation

- Understand that differentiation is the process by which we find the rate at which a quantity is changing. This concept is fundamental to
**calculus**. - Master the basic rules of differentiation such as power rule, quotient rule, product rule and chain rule.

Exponential Differentiation

- Understand the concept of an
**exponential function**. These are functions where a constant base is raised to a variable power(y = a^x

). - Know that the derivative of an exponential function
y = e^x

is simplyy' = e^x

i.e., the derivative of e^x with respect to x is itself. - For other bases (where 'a' is the base, other than e), use the formula
y' = a^x * ln(a)

Logarithmic Differentiation

- Recognise that natural logarithmic functions are inverse of the exponential function with base e (
y = ln x

). - Understand that the derivative of
y = ln x

isy' = 1/x

. - Study how to differentiate logarithmic functions with different bases using the change of base formula.

Differentiating More Complex Functions

- Learn to apply the
**chain rule**to differentiate more complex exponential and logarithmic functions. This rule involves differentiating a function of a function. - Master the application of
**product rule**and**quotient rule**where appropriate.

Practice and Application

- Constantly practice the application of these rules in increasingly complex scenarios.
- Apply these principles in real world problems, such as exponential growth and decay, and elasticity in economics.