Calculus: Derivatives and Exponential Functions

Study Notes

Definition: A derivative measures the rate of change of a function with respect to one of its variables.
Notation: f'(x) or (d/dx)f(x)
Interpretation: The derivative of a function f at a point x represents the rate of change of the function at that point.
Geometric Interpretation: The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

Exponential functions: f(x) = a^x, where 'a' is a positive real number.
Derivative of exponential functions: f'(x) = a^x * ln(a)
Applications:
- Population growth and decay
- Compound interest
- Chemical reactions

Trigonometric functions: sine, cosine, and tangent.
Derivatives of trigonometric functions:
- (d/dx)sin(x) = cos(x)
- (d/dx)cos(x) = -sin(x)
- (d/dx)tan(x) = sec^2(x)
Applications:
- Modeling periodic phenomena (e.g., sound waves, light waves)
- Analyzing circular motion

Local maximum: A point at which the function has a maximum value in a small region around that point.
Local minimum: A point at which the function has a minimum value in a small region around that point.
Critical points: Points at which the derivative of the function is zero or undefined.
First Derivative Test:
- If f'(x) changes sign from positive to negative, then x is a local maximum.
- If f'(x) changes sign from negative to positive, then x is a local minimum.

Concave up: A function is concave up if its derivative is increasing.
Concave down: A function is concave down if its derivative is decreasing.
Inflection points: Points at which the concavity of a function changes.
Second Derivative Test:
- If f''(x) > 0, then the function is concave up.
- If f''(x) < 0, then the function is concave down.

Measures the rate of change of a function with respect to one of its variables
Notated as f'(x) or (d/dx)f(x)
Represents the rate of change of the function at a point
Geometrically, it's the slope of the tangent line to the graph of the function at a point

Defined as f(x) = a^x, where 'a' is a positive real number
Derivative: f'(x) = a^x * ln(a)
Applies to:
- Population growth and decay
- Compound interest
- Chemical reactions

Include sine, cosine, and tangent
Derivatives:
- (d/dx)sin(x) = cos(x)
- (d/dx)cos(x) = -sin(x)
- (d/dx)tan(x) = sec^2(x)
Apply to:
- Modeling periodic phenomena (e.g., sound waves, light waves)
- Analyzing circular motion

Local maximum: A point with the maximum value in a small region
Local minimum: A point with the minimum value in a small region
Critical points: Where the derivative is zero or undefined
First Derivative Test:
- Sign change from positive to negative: local maximum
- Sign change from negative to positive: local minimum