Derivatives: Rules and General Formulas

Basic Derivative Rules

Derivatives are fundamental in calculus, representing the rate of change of a function. The following rules are essential for differentiating various types of functions:

• Power Rule: , for real numbers .

• Sum Rule:

• Difference Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Example: To differentiate , use the chain rule:

• Let ,

Derivatives of Trigonometric, Exponential, Logarithmic, and Hyperbolic Functions

Trigonometric Functions

Inverse Trigonometric Functions

Exponential and Logarithmic Functions

Hyperbolic Functions

Applications of Derivatives

Velocity, Speed, and Acceleration

For an object moving along a line with position :

• Velocity:

• Speed:

• Acceleration:

Finding Tangent Lines

The tangent line to at has slope and passes through . The equation is:

Example: For at :

• Compute using the quotient rule.

• Find and write the tangent line equation.

Implicit Differentiation

Used when cannot be explicitly solved in terms of . Differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

• Apply chain rule and product rule as needed.

• Solve for in terms of and .

Finding Tangent Lines with Implicit Functions

To find the tangent line to a curve defined implicitly, differentiate both sides, solve for , and substitute the point to find the slope.

Logarithmic Differentiation

Logarithmic differentiation is useful for functions involving products, quotients, or powers. Steps:

1. Take the natural logarithm of both sides.

2. Differentiate using chain rule and properties of logarithms.

3. Solve for .

Higher-Order Derivatives

Second and Third Derivatives

Higher-order derivatives are found by repeatedly differentiating the function. The second derivative measures the rate of change of the first derivative, and so on.

Related Rates

Procedure for Solving Related-Rate Problems

Related rates involve finding the rate at which one quantity changes with respect to another, often time. Steps:

1. Read the problem carefully and organize information.

2. Write equations expressing relationships among variables.

3. Differentiate with respect to time.

4. Substitute known values and solve.

5. Check units and reasonableness.

Example: Spreading Oil

If the radius of an oil patch increases at a rate of 30 m/hr, how fast is the area increasing when the radius is 100 m?

• Area:

• Substitute , :

• m/hr

Example: Distance Between Moving Objects

Using the Pythagorean theorem, relate the rates of change of distances between two moving objects:

Special Derivative Techniques

Chain Rule for Powers

When differentiating powers of functions, use the chain rule:

Power Rule Applications

The power rule is used for differentiating functions of the form :

Derivatives Involving Logarithms

For functions involving logarithms, use the chain rule and product rule as needed:

Derivatives Involving Inverse Trigonometric Functions

Apply the chain rule to differentiate inverse trigonometric functions:

Tables and Calculating Derivatives at Points

Using Tables to Find Derivatives

Tables can provide values of functions and their derivatives at specific points. Use these values to compute derivatives of combinations of functions.

Summary Table: Derivative Rules

*Additional info: Academic context and examples were expanded for completeness and clarity.*