BackMA 161 Exam 2 Study Guide: Differentiation and Applications
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Rules of Differentiation
Addition, Multiplication by a Constant, Power Rule
The basic rules of differentiation allow us to compute derivatives efficiently for a wide variety of functions.
Addition Rule: The derivative of a sum is the sum of the derivatives.
Multiplication by a Constant: The derivative of a constant times a function is the constant times the derivative.
Power Rule: For any real number , the derivative of is:
Derivative of : The derivative of the exponential function is:
Product Rule and Quotient Rule
These rules are used when differentiating products or quotients of functions.
Product Rule:
Quotient Rule:
Example: If and , then
Trig Rules
Evaluation of Trig Limits
Limits involving trigonometric functions often use standard limit results and identities.
Key Limit:
Example:
Derivatives of Trig Functions
The derivatives of basic trigonometric functions are fundamental in calculus.
Rates of Change
Position, Velocity, and Acceleration Problems
These problems involve interpreting derivatives as rates of change in physical contexts.
Position Function: describes the position at time .
Velocity: The derivative of position with respect to time.
Acceleration: The derivative of velocity with respect to time.
Example: If , then ,
Chain Rule
Application of Chain Rule to Compositions of Functions
The chain rule is used to differentiate composite functions.
Chain Rule: If , then
Multiple Applications: For nested compositions, apply the chain rule repeatedly.
Combined with Product/Quotient Rule: Sometimes, the chain rule is used together with other rules.
Example:
Implicit Differentiation
Derivatives of Implicit Equations
Implicit differentiation is used when is defined implicitly by an equation involving and .
Procedure: Differentiate both sides of the equation with respect to , treating as a function of .
Example: For , differentiate to get , so
Logarithms
Derivatives of , , and
Logarithmic and exponential functions have specific differentiation rules.
Application of Logarithmic Differentiation
Logarithmic differentiation is useful for functions involving products, quotients, or powers.
Procedure: Take the natural logarithm of both sides, differentiate, and solve for .
Example: For , , so , thus
Inverses
Derivatives of Inverse Trig Functions
Inverse trigonometric functions have characteristic derivatives.
Derivatives of Inverse Functions
If and are inverse functions, the derivative of the inverse is:
Example: If , then , and
Related Rates
Solving Related Rates Problems
Related rates problems involve finding the rate at which one quantity changes in relation to another.
Procedure:
Identify all variables and their rates of change.
Write an equation relating the variables.
Differentiate both sides with respect to time.
Substitute known values and solve for the desired rate.
Example: If a circle's radius increases at 2 cm/s, find the rate at which area increases when cm. , , so cm/s
Summary Table: Differentiation Rules
Function | Derivative |
|---|---|
Additional info: This guide omits maxima and minima (extrema) topics, as specified in the original document.
