뒤로Chain Rule, Higher Order Derivatives, and Applications in Business Calculus
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Chain Rule and Composite Functions
Composite Functions
Composite functions are formed when one function is applied to the result of another function. This is a fundamental concept in calculus, especially when differentiating complex expressions.
Definition: If f and g are functions, the composition is defined as .
Example: For , we can write where and .
The Chain Rule
The chain rule is used to differentiate composite functions. It allows us to find the derivative of a function that is composed of two or more functions.
Formula: The derivative of the composition is given by:
Example: For :
Applying the chain rule:
Practice: Differentiation Using the Chain Rule
Differentiate the following expressions using the chain rule:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Example Solution: For (a) :
Let , so
Equations of Tangent Lines
Finding Tangent Lines
The tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve there.
Formula: The equation of the tangent line at is , where and .
Example: For at :
Find using the chain rule, evaluate at , and substitute into the tangent line formula.
Applications: Revenue Rate of Change
Business Application: Revenue Function
In business calculus, the rate at which revenue changes with respect to the number of items sold is important for decision-making.
Example: If , then the rate of change of revenue when is .
Differentiate using the chain rule:
Evaluate at :
Higher Order Derivatives
Definition and Notation
Higher order derivatives are successive derivatives of a function. They are used to analyze the behavior of functions, such as acceleration in physics or concavity in economics.
First derivative: , ,
Second derivative: , ,
Third derivative: , ,
Fourth derivative: , ,
Practice: Second Derivatives
Find the second derivative of the following:
(a)
(b)
(c)
(d)
(e)
Example Solution: For (a) :
First derivative:
Second derivative:
Physical Applications: Velocity and Acceleration
Position, Velocity, and Acceleration
In physics, the position function describes the location of an object at time . The first derivative gives velocity, and the second derivative gives acceleration.
Velocity:
Acceleration:
Relationship:
Example: If :
Velocity:
Acceleration:
At :
Velocity:
Acceleration:
Summary Table: Derivative Notation
The following table summarizes the notation for higher order derivatives:
Order | Notation | Expression |
|---|---|---|
First | , , | Derivative |
Second | , , | Second derivative |
Third | , , | Third derivative |
Fourth | , , | Fourth derivative |
Additional Practice Problems
Differentiation using the chain rule for various expressions, including powers, roots, and quotients.
Finding tangent lines to curves at specified points.
Computing second derivatives for polynomial and rational functions.
Additional info: These notes cover key Business Calculus topics: differentiation, chain rule, higher order derivatives, tangent lines, and applications to business and physics. Practice problems reinforce understanding and provide exam preparation.
