Calculus: Implicit Differentiation and Product Rule

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