Differentiability

A real-valued function $f(x)$ defined on an open interval $(a, b)$ is said to be differentiable at a point $c \in (a, b)$ if the limit of the difference quotient exists finitely. Mathematically, the derivative $f'(c)$ is defined as: \[ f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} \] Geometrically, the difference quotient $\frac{f(c+h)-f(c)}{h}$ represents the slope of the secant line passing through the points $P(c, f(c))$ and $Q(c+h, f(c+h))$ on the curve. As $h \to 0$, the point $Q$ moves along the curve towards $P$. If the limit exists, the secant line approaches a limiting position, which is defined as the tangent line to the curve at $P$. Therefore, $f'(c)$ represents the exact slope of the tangent line to the curve at $x = c$.

• Left-Hand Derivative (LHD): $f'(c^-) = \lim_{h \to 0^+} \frac{f(c-h) - f(c)}{-h}$
• Right-Hand Derivative (RHD): $f'(c^+) = \lim_{h \to 0^+} \frac{f(c+h) - f(c)}{h}$

A fundamental theorem of calculus establishes that differentiability implies continuity. If a function $f(x)$ is differentiable at $x = c$, it is guaranteed to be continuous at $x = c$. \[ \text{Proof: } \lim_{x \to c} (f(x) - f(c)) = \lim_{x \to c} \left[ \frac{f(x) - f(c)}{x - c} \cdot (x - c) \right] = f'(c) \cdot 0 = 0 \implies \lim_{x \to c} f(x) = f(c) \] However, the converse is strictly false. A function can be perfectly continuous at a point but fail to be differentiable. The classic counterexample is $f(x) = |x|$ at $x = 0$. It forms a continuous V-shape path, but $\text{LHD} = -1 \neq 1 = \text{RHD}$.

• Corners: Sudden shifts in direction where both LHD and RHL exist but are unequal (e.g., $f(x) = |x|$ at $x=0$).
• Cusps: Sharp points where the one-sided derivatives approach opposite infinities ($\infty$ and $-\infty$), such as $f(x) = x^{2/3}$ at $x = 0$.
• Vertical Tangents: Points where both one-sided derivatives approach the same infinity ($\pm\infty$), making the tangent line perfectly vertical (e.g., $f(x) = x^{1/3}$ at $x = 0$).
• Discontinuities: Any break, hole, or jump automatically destroys differentiability.

• Product Rule (Leibnitz Rule): $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$
• Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$
• Chain Rule: For composite maps $y = f(g(x))$, let $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = f'(g(x)) \cdot g'(x)$. This generalizes across multiple nested layers.

• Implicit Differentiation: When equations bind variables implicitly as $f(x, y) = 0$, we differentiate all terms with respect to $x$, treating $y$ as a function of $x$ (applying the chain rule to generate $\frac{dy}{dx}$ terms), and then algebraically isolate $\frac{dy}{dx}$.
• Logarithmic Differentiation: Essential for expressions of the form $y = [f(x)]^{g(x)}$ or massive products. Taking the natural log transforms exponents into products: $\ln y = g(x) \ln[f(x)]$. Differentiating yields $\frac{1}{y}\frac{dy}{dx} = g'(x)\ln[f(x)] + g(x)\frac{f'(x)}{f(x)}$.
• Parametric Differentiation: If $x = f(t)$ and $y = g(t)$, then the derivative is calculated as the ratio of their independent rates of change over the parameter: \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)} \quad (\text{provided } f'(t) \neq 0) \]

Derivatives of inverse trigonometric mappings require strict adherence to standard domain restrictions: \[ \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}} \quad (\text{for } |x| < 1) \] \[ \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}, \quad \frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1+x^2} \quad (\text{for } x \in \mathbb{R}) \]

When extending parametric models to the second derivative $\frac{d^2y}{dx^2}$, a common mistake is simply dividing $\frac{d^2y}{dt^2}$ by $\frac{d^2x}{dt^2}$. The correct approach requires applying the chain rule to the first derivative with respect to $x$: \[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}} \]

• Differentiation of Determinants: To differentiate a determinant, differentiate one row (or column) at a time, keeping the remaining rows (or columns) unaltered, and sum the resulting determinants: \[ \frac{d}{dx} \begin{vmatrix} f_1(x) & g_1(x) \\ f_2(x) & g_2(x) \end{vmatrix} = \begin{vmatrix} f_1'(x) & g_1'(x) \\ f_2(x) & g_2(x) \end{vmatrix} + \begin{vmatrix} f_1(x) & g_1(x) \\ f_2'(x) & g_2'(x) \end{vmatrix} \]
• Leibnitz Theorem for $n$-th Derivatives: Generalizes the product rule to higher-order calculations using binomial coefficients: \[ d^n(uv) = \sum_{k=0}^n \binom{n}{k} u^{(n-k)} v^{(k)} \]

• Rolle's Theorem: Let $f:[a, b] \to \mathbb{R}$ be a function that satisfies:
  1. Continuous on the closed interval $[a, b]$
  2. Differentiable on the open interval $(a, b)$
  3. $f(a) = f(b)$
  Then, there exists at least one real number $c \in (a, b)$ such that $f'(c) = 0$. Geometrically, this means the curve has a tangent line parallel to the x-axis.
• Lagrange's Mean Value Theorem (LMVT): Drops the endpoint equality constraint. If conditions (i) and (ii) hold, there exists at least one $c \in (a, b)$ such that: \[ f'(c) = \frac{f(b) - f(a)}{b - a} \] Geometrically, the tangent line at $x=c$ is parallel to the secant line connecting the endpoints $(a, f(a))$ and $(b, f(b))$.

• Cauchy's Mean Value Theorem: If two functions $f(x)$ and $g(x)$ are continuous on $[a, b]$ and differentiable on $(a, b)$, with $g'(x) \neq 0$ for all $x \in (a, b)$, then there exists at least one $c \in (a, b)$ such that: \[ \frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)} \]
• Taylor's and Maclaurin's Series: If a function has derivatives of all orders throughout a neighborhood of $x=a$, it can be represented as a power series. When expanded around $a=0$, it is called a Maclaurin series: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots + \frac{f^{(n)}(0)}{n!}x^n + \dots \]