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Questions and Answers
Which step is NOT part of the logarithmic differentiation process?
Which step is NOT part of the logarithmic differentiation process?
- Take the natural logarithm of both sides
- Isolate y on one side of the equation (correct)
- Use logarithmic properties to simplify the equation
- Differentiate both sides with respect to t
In rate of change problems for particle motion, what does the acceleration function represent?
In rate of change problems for particle motion, what does the acceleration function represent?
- The rate of change of position
- The position of the particle over time
- The change in velocity over time (correct)
- The total distance traveled by the particle
How is the velocity function derived from the position function?
How is the velocity function derived from the position function?
- By applying the quotient rule
- By solving the equation for t
- By integrating the position function
- By differentiating the position function (correct)
What is the formula for the product rule of differentiation?
What is the formula for the product rule of differentiation?
When applying the quotient rule, which statement is correct?
When applying the quotient rule, which statement is correct?
What does solving v(t) = 0 help to identify in particle motion?
What does solving v(t) = 0 help to identify in particle motion?
When is implicit differentiation most beneficial?
When is implicit differentiation most beneficial?
Which of the following functions would most likely require logarithmic differentiation to differentiate?
Which of the following functions would most likely require logarithmic differentiation to differentiate?
What is the primary purpose of implicit differentiation?
What is the primary purpose of implicit differentiation?
When applying the chain rule for the function $y = -11 ext{cos}(u)$ where $u = e^{4x}$, what is the main step you must not overlook?
When applying the chain rule for the function $y = -11 ext{cos}(u)$ where $u = e^{4x}$, what is the main step you must not overlook?
How do you find the slope of the tangent line at a point $(x_0, y_0)$ using implicit differentiation?
How do you find the slope of the tangent line at a point $(x_0, y_0)$ using implicit differentiation?
What is the derivative of $sin^{-1}(-2x)$ with respect to $x$?
What is the derivative of $sin^{-1}(-2x)$ with respect to $x$?
What is the first step in using method of function tables for derivatives?
What is the first step in using method of function tables for derivatives?
Which of the following correctly expresses the result of applying the chain rule on $y = e^{u}$ where $u$ is a function of $x$?
Which of the following correctly expresses the result of applying the chain rule on $y = e^{u}$ where $u$ is a function of $x$?
In differentiating the equation $x^4y - xy^3 = -2$, what supplementary term needs to be added for each term involving $y$?
In differentiating the equation $x^4y - xy^3 = -2$, what supplementary term needs to be added for each term involving $y$?
Which of the following statements about the derivative of $cos(x)$ is correct?
Which of the following statements about the derivative of $cos(x)$ is correct?
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Study Notes
Logarithmic Differentiation
- A technique used to differentiate complex functions, such as those with products, quotients, or powers of functions.
- Involves taking the natural logarithm of both sides of the equation and using logarithmic properties to simplify.
- Differentiation is then performed implicitly, and the equation is solved for 𝑑𝑦/𝑑𝑡.
Rate of Change and Particle Motion
- Focuses on analyzing a particle's position, velocity, and acceleration.
- The position function 𝑠(𝑡) describes the particle's location at time 𝑡.
- Velocity 𝑣(𝑡) is the derivative of the position function, indicating the speed and direction of the particle's movement.
- Acceleration 𝑎(𝑡) is the derivative of the velocity function, describing the change in velocity over time.
Product Rule and Quotient Rule
- Product Rule: Applies when differentiating the product of two functions: 𝑑/𝑑𝑥(𝑓(𝑥)⋅𝑔(𝑥))=𝑓′(𝑥)⋅𝑔(𝑥)+𝑓(𝑥)⋅𝑔′(𝑥).
- Quotient Rule: Applies when differentiating the quotient of two functions: 𝑑/𝑑𝑥(𝑓(𝑥)/𝑔(𝑥))=(𝑓′(𝑥)⋅𝑔(𝑥)−𝑓(𝑥)⋅𝑔′(𝑥))/(𝑔(𝑥))^2
Implicit Differentiation
- Used when a function is not explicitly solved for 𝑦 in terms of 𝑥.
- Differentiates both sides of the equation with respect to 𝑥, treating 𝑦 as a function of 𝑥.
- Applies the chain rule for terms involving 𝑦, adding a 𝑑𝑦/𝑑𝑥 term for each 𝑦.
Using Function Tables for Derivatives
- Provides values of 𝑓(𝑥), 𝑔(𝑥) and their derivatives 𝑓′(𝑥), 𝑔′(𝑥) in a table format.
- Uses product, quotient, or chain rules to differentiate expressions involving 𝑓(𝑥) and 𝑔(𝑥).
- Plugs in values from the table to compute the derivative at a specified point.
Chain Rule
- Applies to differentiating a composite function 𝑓(𝑔(𝑥)).
- Formula: 𝑑𝑦/𝑑𝑥=𝑑𝑦/𝑑𝑢⋅𝑑𝑢/𝑑𝑥.
Trigonometric Differentiation
- Requires knowledge of derivatives of basic trigonometric functions:
- 𝑑/𝑑𝑥(sin(𝑥))=cos(𝑥),
- 𝑑/𝑑𝑥(cos(𝑥))=−sin(𝑥).
- Also includes derivatives of inverse trigonometric functions, such as:
- 𝑑/𝑑𝑥(sin−1(𝑥))=1/√(1−𝑥^2)
Finding Tangent Lines
- Finds the equation of the tangent line to a curve at a given point (𝑥0, 𝑦0).
- Differentiates the equation to find the slope 𝑚 at the point.
- Uses the point-slope form of a line: 𝑦−𝑦0=𝑚(𝑥−𝑥0)
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