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.