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Questions and Answers
Why is implicit differentiation necessary when dealing with certain types of functions?
Why is implicit differentiation necessary when dealing with certain types of functions?
- It avoids the chain rule in differentiation.
- It simplifies complex algebraic expressions.
- It is only used for polynomial functions.
- It allows finding derivatives of functions not explicitly solved for one variable. (correct)
In the context of implicit differentiation, what does it mean for a relation between $x$ and $y$ to be 'implicit'?
In the context of implicit differentiation, what does it mean for a relation between $x$ and $y$ to be 'implicit'?
- The relation is defined such that $y$ is not explicitly expressed as a function of $x$. (correct)
- The relation is linear.
- The relation is composed of trigonometric functions only.
- The relation can be easily rewritten to isolate $y$ as a function of $x$.
When performing implicit differentiation on an equation, what is the significance of differentiating each term with respect to $x$?
When performing implicit differentiation on an equation, what is the significance of differentiating each term with respect to $x$?
- It treats $y$ as a function of $x$, allowing us to apply the chain rule when differentiating terms involving $y$. (correct)
- It simplifies the equation by treating $y$ as a constant.
- It eliminates the need for the chain rule.
- It is only necessary when the equation is a polynomial.
Consider the equation $x^2 + y^2 = 25$. When finding $\frac{dy}{dx}$ using implicit differentiation, why is it important to apply the chain rule to the $y^2$ term?
Consider the equation $x^2 + y^2 = 25$. When finding $\frac{dy}{dx}$ using implicit differentiation, why is it important to apply the chain rule to the $y^2$ term?
What is the critical step after differentiating both sides of an implicit equation with respect to $x$?
What is the critical step after differentiating both sides of an implicit equation with respect to $x$?
Why is factoring out $\frac{dy}{dx}$ a crucial step in solving for the derivative in implicit differentiation?
Why is factoring out $\frac{dy}{dx}$ a crucial step in solving for the derivative in implicit differentiation?
In implicit differentiation, what does the notation $\frac{d}{dx}$ represent?
In implicit differentiation, what does the notation $\frac{d}{dx}$ represent?
How does implicit differentiation differ from explicit differentiation?
How does implicit differentiation differ from explicit differentiation?
Given the equation $x^2 + \sin(y) = y^3$, which step is essential after applying $\frac{d}{dx}$ to both sides?
Given the equation $x^2 + \sin(y) = y^3$, which step is essential after applying $\frac{d}{dx}$ to both sides?
For the implicit equation $x^3 + y^3 = 6xy$, what rule is essential when differentiating the term $6xy$ with respect to $x$?
For the implicit equation $x^3 + y^3 = 6xy$, what rule is essential when differentiating the term $6xy$ with respect to $x$?
After finding $\frac{dy}{dx}$ using implicit differentiation, how do you determine the slope of the tangent line at a specific point $(a, b)$ on the curve?
After finding $\frac{dy}{dx}$ using implicit differentiation, how do you determine the slope of the tangent line at a specific point $(a, b)$ on the curve?
You've used implicit differentiation to find $\frac{dy}{dx}$ for an equation. The expression for $\frac{dy}{dx}$ still contains both $x$ and $y$. Why is this typically the case?
You've used implicit differentiation to find $\frac{dy}{dx}$ for an equation. The expression for $\frac{dy}{dx}$ still contains both $x$ and $y$. Why is this typically the case?
What is the significance of the derivative $\frac{dy}{dx}$ obtained through implicit differentiation?
What is the significance of the derivative $\frac{dy}{dx}$ obtained through implicit differentiation?
How does implicit differentiation extend the application of differentiation beyond explicitly defined functions?
How does implicit differentiation extend the application of differentiation beyond explicitly defined functions?
Consider the implicit function $\tan(x+y) = x$. What is the next step after applying the derivative with respect to x on both sides?
Consider the implicit function $\tan(x+y) = x$. What is the next step after applying the derivative with respect to x on both sides?
Flashcards
What is an explicit function?
What is an explicit function?
A form where the independent variable is completely isolated.
What is an implicit function?
What is an implicit function?
A relation where the independent variable is not isolated.
What is implicit differentiation?
What is implicit differentiation?
A technique to find dy/dx when y is not explicitly defined.
First step of implicit differentiation?
First step of implicit differentiation?
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Second step of implicit differentiation?
Second step of implicit differentiation?
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Third step of implicit differentiation?
Third step of implicit differentiation?
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Fourth step of implicit differentiation?
Fourth step of implicit differentiation?
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What is the derivative of arccosx?
What is the derivative of arccosx?
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Why use implicit differentiation?
Why use implicit differentiation?
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Study Notes
- Implicit Differentiation in Basic Calculus is for Senior High School, Module 17, Quarter 3.
Implicit Differentiation
- Functions are not always easily written with the independent variable isolated from the dependent variable.
- Some relations may make it impossible to isolate variables easily.
- Functions and relations like these are called implicit, as in the following equations: x³ + y³ + 4xy = 0, y² – 2x = 0, or x² + y² - 36 = 0
- An implicit relation means that a value x determines one or more values of y, even without a simple formula for the y values.
- Previous differentiation examples dealt with equations of the form y = f(x), where y is explicitly defined in terms of x.
- When you are unable to solve for y as a function of x, like in x2 – 3y3 + 5y = 7, previous differentiation rules may not work.
- To find dy/dx for a given equation, Implicit Differentiation will be used.
Steps in Performing Implicit Differentiation
- Differentiate both sides of the equation with respect to x.
- Collect all terms involving dy/dx on the left side of the equation, and move all other terms to the right side of the equation.
- Factor dy/dx out of the left side of the equation.
- Solve for dy/dx.
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