100. Derivatives
Measures the instantaneous rate of change of a function
Derivative of of a function
General Pattern:
- Linear term: If a function is linear (e.g., ), the first derivative will be a constant .
- Constant term: The derivative of a constant is zero.
- Higher-degree polynomials: For polynomials of higher degrees, each derivative decreases the degree of the polynomial until you get a constant, after which all further derivatives are zero.
Example
Consider the quartic function:
- First derivative: (cubic)
- Second derivative: (quadratic)
- Third derivative: (linear)
- Fourth derivative: (constant)
- Fourth derivative: (zero)
100.0.1. Power Rule
Example
100.0.2. Product Rule
Example
- Compute the derivatives of and
- Apply rule
100.0.3. Quotient Rule
Example
- Compute the derivatives of and
- Apply rule
100.0.4. Chain Rule
Example
- Outer function:
- Inner function
- Compute the derivatives of and
- Apply rule
100.0.5. Others
1. Trigonometric Functions
2. Inverse Trigonometric Functions
3. Exponential and Logarithmic Functions
Definition 0: Stationary Point
A stationary point of a function is a solution to That is, where the slope (derivative) is zero.