1.10.1: Resources and Key Concepts

Resources

Videos

• Derivative function
  • The Derivative as a Function - The Derivative Function
• Differentiability
  • The Derivative as a Function - Differentiability: To Be or Not To Be
• Notation for derivatives
  • The Derivative as a Function - Derivative Notations
• Graphing derivatives
  • The Derivative as a Function - Graphing Derivatives and Finding Patterns
• Differentiability and continuity
  • Proofs in Differential Calculus - Differentiability Implies Continuity
• Higher-order derivatives
  • The Derivative as a Function - Higher-Order Derivatives

Key Concepts

Definitions

• Derivative Function: Let \(f\) be a function. The derivative function, denoted by \(f^{\prime}\), is the function whose domain consists of those values of \(x\) such that the following limit exists: \(f^{\prime}(x) = \displaystyle \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\).
• Differentiable at \(a\): A function \(f(x)\) is said to be differentiable at \(a\) if \(f^{\prime}(a)\) exists.
• Differentiable on \(S\): A function is said to be differentiable on \(S\) if it is differentiable at every point in an open set \(S\).
• Differentiable Function: A differentiable function is one in which \(f^{\prime}(x)\) exists on its domain.
• Lagrange's Notation (for derivative): \(f^{\prime}(x)\) or \(y^{\prime}\). If evaluated at \(x=a\), it's \(f^{\prime}(a)\) or \(y^{\prime}(a)\).
• Leibniz's Notation (for derivative): \(\frac{dy}{dx}\). Other forms include \(\frac{df(x)}{dx}\), \(\frac{df}{dx}\), \(\frac{d}{dx}f(x)\). If evaluated at \(x=a\), it's \(\left.\frac{dy}{dx}\right|_{x=a}\) or \(\left.\frac{df}{dx}\right|_{x=a}\).
• Euler's Notation (for derivative): \((Df)(x)\), using the differential operator \(D\).
• Newton's Notation (for derivative): \(\dot{y}\) (if \(y\) is a function of \(t\)).
• Second Derivative: The derivative of the derivative function.
• Higher-Order Derivatives: Derivatives beyond the first derivative (e.g., second, third, fourth, \(n^{th}\) derivatives). Notations include \(f^{\prime\prime}(x), f^{\prime\prime\prime}(x), f^{(n)}(x)\), \(y^{\prime\prime}, y^{\prime\prime\prime}, y^{(n)}\), \(\frac{d^2y}{dx^2}, \frac{d^ny}{dx^n}\).

Theorems

• Theorem: Differentiability Implies Continuity: Let \(f(x)\) be a function and \(a\) be in its domain. If \(f(x)\) is differentiable at \(a\), then \(f\) is continuous at \(a\).

Common Mistakes

• Careful Use of Leibniz's Notation: When using Leibniz's notation to find the derivative of \(y=f(x)\) at \(x=a\), it is incorrect to write \(\frac{df(a)}{dx}\). The correct forms are \(\left.\frac{dy}{dx}\right|_{x=a}\) or \(\left.\frac{df}{dx}\right|_{x=a}\).
• Continuity Does Not Imply Differentiability: A function can be continuous at a point but not differentiable there. Reasons include:
  • A sharp corner (e.g., \(f(x)=|x|\) at \(x=0\)).
  • A vertical tangent line (e.g., \(f(x)=\sqrt[3]{x}\) at \(x=0\)).
  • Oscillating behavior (e.g., \(f(x) = x\sin(1/x)\) for \(x \neq 0\), \(f(0)=0\), at \(x=0\)).