6 Questions
If $y = f(u)$ and $u = g(x)$, what is the formula for finding $rac{dy}{dx}$ using the chain rule?
$rac{dy}{dx} = rac{dy}{du} imes rac{du}{dx}$
If $y = u imes v$, what is the formula for finding $rac{dy}{dx}$ using the product rule?
$rac{dy}{dx} = u imes rac{dv}{dx} + v imes rac{du}{dx}$
If $y = f(x)$, how would you find $rac{d^2 y}{dx^2}$ using implicit differentiation?
$rac{d^2 y}{dx^2} = rac{d}{dx} ig(rac{dy}{dx}ig)$
If $y = rac{u}{v}$, what is the formula for finding $rac{dy}{dx}$ using the quotient rule?
$rac{dy}{dx} = rac{v imes rac{du}{dx} - u imes rac{dv}{dx}}{v^2}$
What is the formula for finding $rac{dy}{dx}$ for an implicit function $F(x, y) = 0$?
$rac{dy}{dx} = - rac{rac{ ext{d} F}{ ext{d} x}}{rac{ ext{d} F}{ ext{d} y}}$
If $F(x, y) = x^2 + y^2 - 25 = 0$, find $rac{dy}{dx}$ using implicit differentiation.
$rac{dy}{dx} = - rac{2x}{2y} = - rac{x}{y}$
Study Notes
Implicit Differentiation
Chain Rule
- Used to find the derivative of a composite function
- Formula: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
- Example: If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
Product Rule
- Used to find the derivative of a product of functions
- Formula: $\frac{d}{dx} (u \cdot v) = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx}$
- Example: If $y = u \cdot v$, then $\frac{dy}{dx} = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx}$
Higher-order Derivatives
- Implicit differentiation can be used to find higher-order derivatives
- Formula: $\frac{d^n y}{dx^n} = \frac{d}{dx} \left(\frac{d^{n-1} y}{dx^{n-1}}\right)$
- Example: If $y = f(x)$, then $\frac{d^2 y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right)$
Quotient Rule
- Used to find the derivative of a quotient of functions
- Formula: $\frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}$
- Example: If $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}$
Implicit Functions
- An implicit function is a function that is defined implicitly by an equation
- Example: $x^2 + y^2 - 25 = 0$
- Implicit differentiation is used to find the derivative of an implicit function
- Formula: $\frac{dy}{dx} = - \frac{\partial F/\partial x}{\partial F/\partial y}$
- Example: If $F(x, y) = x^2 + y^2 - 25 = 0$, then $\frac{dy}{dx} = - \frac{2x}{2y} = - \frac{x}{y}$
Implicit Differentiation
Chain Rule
- Used to find the derivative of a composite function
- Formula involves derivative of outer function and inner function
- Applied when function is a composition of two functions, e.g., y = f(u) and u = g(x)
Product Rule
- Used to find the derivative of a product of functions
- Formula involves derivative of each function and their product
- Applied when function is a product of two functions, e.g., y = u * v
Higher-order Derivatives
- Implicit differentiation can be used to find higher-order derivatives
- Formula involves repeated differentiation of previous derivative
- Enables finding higher-order derivatives of a function, e.g., second derivative of y = f(x)
Quotient Rule
- Used to find the derivative of a quotient of functions
- Formula involves derivative of numerator and denominator, and their product
- Applied when function is a quotient of two functions, e.g., y = u/v
Implicit Functions
- An implicit function is a function defined implicitly by an equation
- Example: x^2 + y^2 - 25 = 0
- Implicit differentiation is used to find the derivative of an implicit function
- Formula involves partial derivatives of the implicit function with respect to x and y
- Enables finding the derivative of an implicit function, e.g., dy/dx = -x/y
Implicit differentiation and product rule are important concepts in calculus. This quiz covers the formulas and examples of chain rule and product rule.
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