BackCalculus Study Guide: Derivatives, Rules, and Applications
Study Guide - Smart Notes
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Power Rule and Its Variations
Standard Power Rule
The power rule is a fundamental technique for finding derivatives when the base is the variable x.
Formula:
Application: Used for differentiating polynomials and any function where the variable is raised to a constant power.
Example:
Power Rule with Variable Base and Chain Rule
When the base is a function of x (not just x itself), the chain rule must be applied.
Formula:
Application: Used for composite functions where the inner function is not simply x.
Example:
Derivatives of Trigonometric Functions
Basic Trigonometric Derivatives
There are six primary trigonometric functions, each with a standard derivative.
Trigonometric Derivatives with Chain Rule
When the argument of the trigonometric function is itself a function of x, use the chain rule.
Product and Quotient Rules
Product Rule
The product rule is used to differentiate the product of two functions.
Formula:
Example:
Quotient Rule
The quotient rule is used to differentiate the ratio of two functions.
Formula:
Example:
L'Hôpital's Rule
Indeterminate Forms and Limits
L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms such as or .
Formula: (if the original limit is indeterminate)
Application: Differentiate numerator and denominator separately, then take the limit.
Example:
Physics Applications: Motion and Acceleration
Distance, Velocity, and Acceleration
Calculus is used to relate position, velocity, and acceleration in physics.
Distance (s): The position of an object as a function of time.
Velocity (v): The derivative of position with respect to time:
Acceleration (a): The derivative of velocity with respect to time:
Example: If , then ,
Projectile Motion
For objects in free fall or projectile motion, the following equations are commonly used:
Where: is initial height, is initial velocity, is acceleration due to gravity.
Pythagorean Identities and Reciprocal Identities
Pythagorean Identities
These identities relate the squares of sine, cosine, and other trigonometric functions.
Identity | Equation |
|---|---|
Basic | |
Secant-Tangent | |
Cosecant-Cotangent |
Reciprocal Identities
These identities express trigonometric functions as reciprocals of each other.
Function | Reciprocal |
|---|---|
Double Angle Identities
Formulas for Double Angles
Double angle identities are used to simplify expressions involving trigonometric functions of .
Equation of a Line
Point-Slope and Slope-Intercept Forms
The equation of a tangent line to a function at a given point can be found using the derivative.
Point-Slope Form: , where is the slope at
Slope-Intercept Form:
Application: Use the derivative to find the slope at a point, then plug into the point-slope formula.
Example: For at , slope , so
Practice Problems (Summary)
Types of Problems
The study guide suggests practice problems involving:
Finding derivatives of functions
Using the tangent line equation
Applying the point-slope formula
Solving for the slope and y-intercept
Additional info: The guide references finding the equation of a tangent line and using derivatives for motion problems, which are standard applications in Calculus I.
