Derivatives and Their Rules

Basic Derivative Rules (Section 3.3)

The derivative measures the instantaneous rate of change of a function. Calculating derivatives is fundamental in calculus, and several rules simplify the process.

• Power Rule: For , the derivative is .

• Sum Rule: The derivative of a sum is the sum of the derivatives: .

• Constant Multiple Rule: , where is a constant.

• Finding Tangent Slopes: To find where the tangent has a specific slope , solve for .

• Higher-Order Derivatives: The second derivative measures the rate of change of the first derivative; higher-order derivatives follow similarly.

• Example: For , , .

Product and Quotient Rules (Section 3.4)

When differentiating products or quotients of functions, specialized rules are used.

• Product Rule:

• Quotient Rule:

• Instantaneous Growth Rate: The derivative gives the instantaneous rate of change at .

• Steady-State Population: In population models, steady-state occurs where .

• Example: For , .

Derivatives of Trigonometric Functions (Section 3.5)

Trigonometric functions have well-defined derivatives, which are essential in many calculus problems.

• Basic Derivatives:

• Higher-Order Derivatives: Repeated differentiation of trigonometric functions often cycles through the original function and its negatives.

• Example:

Advanced Differentiation Techniques

Chain Rule (Section 3.7)

The chain rule is used to differentiate composite functions. There are two main forms of the chain rule.

• Standard Form: If , then

• Theorem 3.12 (Alternative Form): If and , then

• Example: For ,

Implicit Differentiation (Section 3.8)

Implicit differentiation is used when functions are defined implicitly rather than explicitly.

• Implicit Differentiation: Differentiate both sides of the equation with respect to , treating as a function of $x$.

• Tangent Line Equation: At point , the tangent line is , where at $(x_0, y_0)$.

• Second Derivative: After finding , differentiate again to find , using implicit differentiation.

• Example: For ,

Related Rates (Section 3.9)

Related rates problems involve finding the rate at which one quantity changes in relation to another.

• Procedure:

  1. Identify all variables and their rates of change.

  2. Write an equation relating the variables.

  3. Differentiate both sides with respect to time .

  4. Substitute known values and solve for the unknown rate.

• Example: If a circle's radius increases at 2 cm/s, the area increases at .

Applications of the Derivative

Extreme Values and Critical Points (Section 4.1)

Derivatives are used to identify extreme values (maximums and minimums) and critical points of functions.

• Critical Points: Points where or is undefined.

• Absolute Extreme Values: The highest or lowest value of on a given interval.

• Local Extreme Values: Maximum or minimum values in a neighborhood around a point.

• Procedure:

  1. Find and solve for critical points.

  2. Evaluate at critical points and endpoints (if interval is closed).

  3. Compare values to determine absolute and local extremes.

• Example: For , critical point at , .