Continuity and Differentiability - Second order derivatives

Definition of Second Order Derivative: If a function $y = f(x)$ is differentiable, its derivative $\frac{dy}{dx}$ is also a function of $x$. If $\frac{dy}{dx}$ is differentiable again, its derivative is called the second order derivative of $y$ with respect to $x$, denoted as $\frac{d^2y}{dx^2}$ or $f''(x)$. Visually, this represents the rate of change of the slope of the tangent to the curve.

Geometric Interpretation and Concavity: The second derivative describes the curvature of the graph. If $f''(x) > 0$ on an interval, the slope is increasing, and the graph is concave upwards, forming a 'cup' or 'U' shape. If $f''(x) < 0$, the slope is decreasing, and the graph is concave downwards, forming a 'cap' or inverted 'U' shape.

Notation Variants: In CBSE problems, the second order derivative is represented using various symbols such as $y_2$, $y''$, $f''(x)$, or $D^2y$. The notation $y_1$ and $y_2$ is particularly common in problems requiring the verification of differential equations, where $y_1 = \frac{dy}{dx}$ and $y_2 = \frac{d^2y}{dx^2}$.

Parametric Differentiation: For functions defined by $x = f(t)$ and $y = g(t)$, the second derivative $\frac{d^2y}{dx^2}$ is found by differentiating the first derivative $\frac{dy}{dx}$ with respect to $t$ and then multiplying by $\frac{dt}{dx}$. It is expressed as $\frac{d^2y}{dx^2} = \frac{d}{dt}(\frac{dy}{dx}) \cdot \frac{dt}{dx}$. Visually, this accounts for the fact that the parameter $t$ is the intermediate variable for both $x$ and $y$.

Points of Inflection: A point on a curve where the concavity changes (e.g., from concave up to concave down) is called a point of inflection. At such a point, $f''(x) = 0$ or is undefined. On a graph, this is the transition point between the 'cup' and 'cap' sections of the curve.

Successive Differentiation in Identities: Many exam problems involve proving an identity such as $a y_2 + b y_1 + cy = 0$. This often requires calculating $y_1$ and $y_2$, substituting them into the left-hand side of the equation, and simplifying to show it equals zero.

Physical Significance: If $s = f(t)$ represents the displacement of an object at time $t$, then the first derivative $\frac{ds}{dt}$ represents velocity ($v$), and the second order derivative $\frac{d^2s}{dt^2}$ represents the acceleration ($a$) of the object.